Now, I have limited understanding of this, and there's one main source that I base this answer on, which are Nikita Khrushchev's memoirs, hopefully it will be adequate. I've even asked a question about any sources backing up or refuting Khrushchev on this, but that has not been answered.

By the mid 1930s it had become clear to the soviets that war was approaching, in their minds against the antisoviet foreign powers, the' imperialists'. By 1934, war readiness was a central idea in bolshevik politics, with Stalin's opening speech on the party congress of that year calling for readiness for war, citing Japanese expansion and the rise of German fascism among other threats.

It is undeniable also that despite the non-aggression pact between the soviet union and nazi germany, both sides knew that war between them was inevitable. For this reason, the soviets had been developing the red army at a rapid rate, also rejuvenating their purged military personnel after initial failures during the winter war. So the soviets knew that war was coming, still, why did they not expect the German attack?

This issue boils down to several disputed possibilities, all of them having the same result: Stalin simply did not expect operation barbarossa and, being in total control of virtually everything in the USSR, him being caught off guard meant the entire USSR would not be ready. The question of why Stalin didn't expect barbarossa is one with disputed answers. There are people who have written on Stalin (e.g. Edvard Radzinsky) who state that Stalin didn't expect barbarossa due to its undeniable and incredible stupidity. It would open up a second front to a war that had already devolved into a stalemate with the nazis being incapable of properly invading England. Others state that for some inexplicable reason, Stalin trusted the non-aggression pact. Either way, we know that even after being repeatedly warned about operation Barbarossa, even being given its exact date, Stalin refused to authorize any hostile move towards Germany.

So that explains why the initial defense was so incompetent, but why the 300K+ figures being captured? Here we can turn to our dear friend Khrushchev to explain for us. To understand what his memoirs have to tell us, however, we have to know two important pieces of information.

First, as a rampup to what the soviets saw as inevitable war, state propaganda began producing pieces focused around war readiness. According both to Khrushchev and propaganda that survives from that time, the important slogans for us to know were "not a step back" and "the war will be fought on in the lands of the enemy". Essentially, state propaganda painted invasion as something that wouldn't happen, would not be allowed to happen, etc. etc.

Second, the "leading figure" of the red army at this time was Kliment Voroshilov. Voroshilov was Stalin's 'military guy' during the great purge: like Kaganovich and Molotov in other fields like art and economics, Voroshilov was the one to authorize military execution lists. He was therefore Stalin's most trusted man in the military. He was also a notoriously bad commander. He was emotionally driven but had battle experience, neither of which he could in the slightest apply to large scale tactics: his leadership was a primary reason for the failures in the early stages of the winter war. Nevertheless, he wasn't removed from active military command until after the beginning of the war with nazi Germany.

These two factors are important because, according to Khrushchev, Voroshilov and Stalin, the two most influential men for the red army, took the propaganda slogans being printed in their name far too seriously. So seriously, in fact, that most of the active personnel in the red army, as well as a lot of military stockpiles, were stationed on the border with Germany, Hungary, and Romania, all of whom would take part in barbarossa. The large surrender of forces you talk about happened in Kiev and Minsk, both part of this heavily armed border region that Khrushchev talks about. So, to answer your question: the soviets armed the German border heavily, then failed to respond in time to the invasion. This led to the capture of massive amounts of red army troops and equipment.

Edit: To expand on the answer because I saw you had more questions than why the large numbers.

As I stated, the large surrenders were mainly at Minsk and Kiev.

Barbarossa was, as I said, unexpected, the response was delayed. The German advance was extremely fast, and both cities were quickly encircled, cutting off supply and essentially guaranteeing that if the soldiers stationed there would not surrender, they would either be killed or die due to a lack of supplies. Not surrendering was suicide.

As for the German soldiers not fighting to the death, as ordered, the answer is simple: people prefer to live and very often don't believe in the ideology that they're forced to fight for. German soldiers, like Russians, surrendered because they wanted to live, and Hitler's orders could do nothing to change that.

Broadly, Soviet soldiers surrendered during Barbarossa for the same reasons that soldiers have always surrendered. Loss of morale; inability to keep fighting because they were cold, or hungry, or had run out of ammunition; inability to keep fighting effectively because of breakdowns in command and unit cohesion, loss of artillery support, or being pinned down by enemy fire; the basic human desire not to die. Put yourself in their shoes - you've volunteered or more likely been conscripted, half of the buddies you made during training have been shot or blown up or have already ditched their uniforms and deserted, you haven't gotten new rations in three days and you were so hungry this morning that you ate the worm in your bread, and every time you see a plane overhead it belongs to the enemy. Sure, there are a lot of your buddies stuck in this town, but they're all going through the same mental process you are. Without effective command, good morale, and adequate supplies , a mass of 100,000 soldiers surrounded in a town is not an effective fighting force.

The Red Army suffered from several factors that exacerbated all of these behaviors. At the materiel and institutional level, they were simply not ready for war (incidentally, this is the explanation I find most compelling for Stalin's seemingly idiotic attempts to stave off war with Germany, going so far as to ignore dozens of separate intelligence reports indicating that a German invasion was obviously imminent). As David Glantz comprehensively demonstrates in Stumbling Colossus: The Red Army on the Eve of World War, the Red Army was understrength in virtually every possible category. Corps commanders lacked adequate radios for their headquarters, artillery units didn't have enough shells for sustained fire, and tank divisions had far fewer operational tanks than their paper strength suggested. This was partly the result of the Soviet Union's incomplete industrialization in 1941; the Five Year Plans had done much to bring the Soviet Union up to speed with Western Europe, but the war industries were not yet able to meet their quotas. This unreadiness was also the result of the Purges and the subsequent military reforms. The Purges absolutely decimated the Red Army officer corps; lieutenants were suddenly promoted to colonels because everyone in the intermediate ranks had been shot or jailed, and the intellectual backbone of the high command of the 1930s had been entirely liquidated. This situation produced poor results in the Winter War, which in conjunction with the shocking success of the Nazi invasion of France urged very hasty reforms to restore much of the suppressed doctrine and force structure; however, between the industrial lag and the structural reshuffling, the Red Army itself did not expect to be ready for war with Germany until 1942 or 1943. Summer 1941 was a very vulnerable time for the Soviets, and this institutional disorder certainly contributed as much as the materiel unreadiness did to the chaos of 1941-42. It is hard to ask men to fight rather than surrender when they lack the material and tactical means to fight effectively.

The morale and political investment of Red Army soldiers is another factor that must shoulder some of the blame for the mass surrender incidents. Elite units like paratroopers drew heavily from Communist Party membership and should be considered to have a high level of political buy-in; however, the Red Army as a whole was a rather mixed bag on this front. One example presented itself in the recruits from the Ukraine and Baltic states: “The fact that, during the prewar years, there were a great number of Red Army men, as well as military service men from the reserves of the western regions of Ukraine and Moldova and the former bourgeois Baltic states...who required additional attention from all political organs.” These recruits, especially those less educated, did not hold the desired level of socialist idealism. During the dark days of late 1941, many an NKVD officer's reports to headquarters mentioned pessimistic grumblings from the troops. Self-mutilations or outright desertions were frequent. Especially before it became apparent how poorly POWs would be treated, many men (especially but I must emphasize not exclusively Ukrainians and members of other nationalities who often felt little loyalty to the Russian-dominated Soviet state) found it preferable to surrender than fight for a cause they held little love for.

Sources & recommended reading:

Glantz, David M. Stumbling Colossus: The Red Army on the Eve of World War. Lawrence, KS: University Press Of Kansas, 2011.

--- and Jonathan M. House. When Titans Clashed: How The Red Army Stopped Hitler. Lawrence, KS: University Press of Kansas, 2015.

Merridale, Catherine. Ivan’s War: Life and Death in the Red Army, 1939-1945. New York: Picador, 2006