All I can get from Wikipedia is "during this Aliyah or phase X number of Jews were absorbed" with no real how they were able to do so. Also, apart from some brief info on Kibbutzim and a bunch of Jews moving to what would become Tel Aviv, the information on the actual development of the country itself is rather scant.
The absorption of the immigrations of each Aliyah was exceedingly difficult at the start, but got better as time went on.
The First Aliyah, composed mainly of immigrants from Russia, comprised about 25,000-30,000 people. They were mostly religiously motivated Zionists (if at all Zionist in the political sense we know it), and the wave lasted from 1881-1903. Most of the immigrants would later pack up and leave due to difficulties in settling, heading back to Russia or to the West. One early settler wrote of his fellow settlers in 1885:
Nothing frightened them, nothing stopped them, neither the barrenness of the country, nor the wildness of the Arabs...nor ignorance of the local language and customs...Nobody knows of all the hardships, sickness, and wretchedness they underwent. No observer from afar can feel what it is like to be without a drop of water for days, to lie for months in cramped tents visited by all sorts of reptiles, or understand what our wives, children, and mothers go through when the Arabs attack us...No one looking at a completed building realizes the sacrifice put into it.
Nearly two dozen settlements were set up, almost all facing immediate financial difficulty and strife. However, only around 3,000 settled in these villages, most entering cities or already established areas. The wealthy Baron Edmond de Rothschild and Moses Montefiore, Zionists themselves, were the main reason the settlements didn't immediately fall apart. Rothschild gave over 1.6 million pounds sterling to the settlers between 1883-1889 to keep them afloat. However, the settlements mostly failed and were in heavy decline by the time Herzl's World Congress began entering the scene in 1897, and the movement (Hibbat Zion) that had launched the first Aliyah eventually joined Herzl's Zionist organization.
The next wave of emigration to Palestine, between 1903-1914, is the Second Aliyah. Precipitated by the pogroms in Russia mainly, the main newcomers were dominated by a socialist ideology and a nationalist fervor, more political than religious in nature. They based their principle on setting up a society revolving around Jewish labor, attempting to exclude Arabs from their economy to fulfill their socialist vision and prove economic independence. This vision would eventually succeed, more or less, and David Ben-Gurion (Israel's first and likely most remembered prime minister) was among the 20,000 or so immigrants who came in this wave. The first Aliyah had established moshavot, a privately owned agricultural settlement. Alongside these, the second Aliyah established kibbutzim, the communally owned agricultural settlements most associate with Israeli agriculture. The immigrants split into two groups: Hapoel Hatzair (The Young Worker) and Poale Zion (Workers of Zion). Both of these groups, dedicated to settlement, agriculture, and a socialist lifestyle, were helped by the established Jewish National Fund, which bought land for them and funneled donations to them. By 1914, there were 44 Jewish agricultural settlements. The World Zionist Organization helped the settlements along through mostly foreign donors, usually Jews in the Diaspora.
The Third Aliyah, carrying some 40,000 Jews, also came mostly from Russia or Eastern Europe. From 1919 to 1923 the immigrants came, once more mostly socialists and young, penniless workers enthusiastic about Zionism. The wave inspired Arab opposition, and rivalries among factions of these workers led to Arab-Jewish strife as well. It was during this Aliyah that Hapoel Hatzair and Poale Zion changed. Poale Zion reconstituted itself as Ahdut Ha'avodah (Unity of Labor), and both parties encouraged Jews to come to Palestine during this wave. In December 1920 the Histadrut, arguably to become the most powerful group in Israel by 1948, was formed as a worker's union in which both Hapoel Hatzair and Ahdut Ha'avodah cooperated as separate entities within the organization, meant to organize Jewish labor. Eventually Ahdut Ha'avodah gained the majority of power over the Histadrut, using it to try and encourage all Jewish businesses to use only Jewish labor, helping to absorb and create jobs for Jewish immigrants. Until the end of the Third Aliyah it didn't seem to be that there were any more than 90,000 Jews in all of Palestine, which had not been too hard to absorb.
The Fourth Aliyah led to another 80,000 or so Jewish immigrants in Palestine, almost all handled once more in the same way as before. From 1924-1928 the Jewish organizations continued buying land, encouraging Jewish business growth, and investing in Jewish businesses in general. It's important to note that there may have been as many as 100,000 people who came to Palestine during this time, but a good many left: in 1927 there were 2,000 more people leaving Palestine than entering, because of a severe economic crisis that led to mass unemployment. Still, Jewish landholdings almost doubled between 1920 and 1929, and the 40 million pounds sterling contributed from Jews abroad between 1917-1929 allowed the agricultural settlements to more than double to 110. The Hebrew University, first of the Israeli universities, was opened in 1925. Despite the troubles economically, Jewish society continued to persevere and grow.
From 1929 to 1939 there is the Fifth Aliyah, within which arrived up to 300,000 Jews (many from places like Germany, where they fled en masse in 1935, over 100,000 during that year alone). Once more, they were absorbed by Jewish economy, which was flourishing especially in the industrial sector. The Jewish groups had managed to secure a 50-50 split on public works sector between Jews and Arabs, despite the population distribution being more than 70% Arab and 30% Jewish, helping the Jewish economy along. The Jewish economy also absorbed many immigrants not only through foreign donors, but also through the immigrants themselves. Between 1933 and 1935 the Jewish population doubled, but most of the people coming were middle-class and brought an infusion of capital with them for the Jewish economy. As a result, industrial firms and cities both grew extremely rapidly, from 6,000 firms in 1930 to 14,000 in 1937, and Jewish economic output would soon outrank Arab output in all but citrus cultivation by 1943. The Jews were also aided greatly by the Arab revolt, during which Arabs went on strike and there was a shortage of vegetables and delays in construction. The jobs of strikers were quickly filled by Jews, increasing Jewish labor's hold, and the closure of the Jaffa port led to the establishment of a port in Tel Aviv instead. Where the Arabs lost out during the revolt, the Jews ended up gaining and taking their place.
From then on, known as Aliyah Bet, another significant amount of Jews (70,000 illegally between 1945 and 1948, many more did not make it or attempt it during WWII, or were caught) entered. The White Paper of 1939 had limited Jewish immigration to 75,000 over a period of 5 years, but coincided with WWII. By halfway through the White Paper, only 35,000 Jewish immigrants had come to Palestine. By 1944, the number was still 20,000 short: the war had severely curtailed Jewish immigration. But the Jewish immigrants during this time would be rather easily absorbed.
Finally, after the war in 1948, the biggest immigration wave would be the one coming between 1948 and 1952. Most of the immigrants came as displaced persons from Europe who were displaced from WWII, or from Arab countries (close to half and half by some estimates, but usually the number is more tilted towards Arab countries providing the most immigrants). This was one of the toughest times for Israel (or the Jewish history in Palestine) as far as immigration: despite having over 600,000 Jews in Israel, the population pretty much doubled with another estimated 680,000 Jews arriving. This was by and far away one of the more difficult tasks the new Israeli government had to tackle, but it did so through a mixture of methods:
British laws, like emergency defense laws, were adopted by Israel and used to great effect. These, like the absentee land laws, allowed Israel to seize land belonging to Arabs who had fled (or some who were still in Israel) their property, and give it to new Jewish immigrants. This was fairly crucial in getting many Jews homes.
Israel looked to foreign donors for help once more. Golda Meir went on a tour of the United States (and met with other Diaspora Jews) to seek donations on behalf of Israel, and asked for help to build 30,000 housing units to hold the refugees. By October 1950, despite the generosity of Western Jews, only 1/3 of the units had been built. A large amount of the funds went instead to building emergency huts for the fast-arriving immigrants. The metal huts kept out the rain, but were furnaces during the summer, and even just putting up wooden frames and nailing canvases to them became a way to house tens of thousands of newcomers. Even a $100 million bank loan did not suffice in helping settle all the newcomers. Vast public works programs were initiated, philanthropy was heavily required, and every effort was made by the new government to house any Jewish immigrant, who it refused to turn away.
It was a tough job, and a tough time, but the immigrants persevered, schools and clinics and businesses were built, and Gilbert writes:
Golda Meir [later to be Israeli Prime Minister during the 1973 Yom Kippur War] once remarked that Israel's propaganda mistake was to do away with its own refugee problem, by treating the refugees as human beings and making every effort to give them a decent home and jobs, The world looked with far greater sympathy on refugees who were kept in their camps (as happened to the Palestinian Arabs in Lebanon, Syria, Egypt and Jordan) and given the minimum of help, or even no help, to integrate into society, The ma'abarot [Jewish refugee camps] had disappeared within a decade; many of the Palestinian: Arab refugee camps remain to this day.