There are a few reasons that the Australasian colonies decided to federate – and the emphasis will differ depending on which (historical) person you ask.

As early as 1857, a Select Committee in the recently formed colony of Victoria wrote:

Your Committe are unanimous in believing that the interest and honor of these growing States would be promoted by the establishment of a system of mutual action and co-operation among them. Their interest suffers, and must continue to suffer, while competing tariffs, naturalization laws, and land systems, rival schemes of immigration, and of ocean postage [...] exist; and the honor and importance which constitute so essential an element of national prosperity, and the absence of which invites aggression from foreign enemies, cannot in this generation belong to any single Colony of the Southern Group; but may, we are persuaded would, be speedily attained by an Australian Federation representing the entire.

[...] By becoming confederates so early in their career, the Australian Colonies would, we believe, immensely economize their strength and resources. [...] They would not only save time and money, but attain increased vigor and accuracy, by treating the larger questions of public policy at one time and place.

[Report from the Select Committee upon the Federal Union of the Australian Colonies, 1856-7]

Note the various reasons listed for federating: tariffs, citizenships, immigration, postage, defense, efficiency.

In 1870, Charles Gavan Duffy spoke in the Victorian Parliament about this previous Select Committee report. His main point was that there had been no action on this report for too long. In pushing for action, he said:

It may be said, no doubt, that England is mistress of the seas, and will be able to protect her commerce and ours. But France and America have been making enormous expenditure and immense exertions for years past to be in a position to compete for this supremacy. Even if it be admitted that England would be able to protect the great highway to Europe by the Cape, will she be able to guard the Northern Pacific, or to save the great Australian cities from fleets stationed at San Francisco or New Caledonia?

[...] it would put an end to what a Canadian statesman describes as “colonies cutting each others throats with razors called tariffs”. It would create between us an intercourse of mind. [...] It would result in the creation of a national spirit [...] And, finally, it would give Australia complete control of her own resources for the protection of her own interests.

Alfred Deakin, an eyewitness to, and key participant in, the process of federation, wrote in his ‘The Federal Story’ (based on notes he kept during the years that federation was discussed and progressed):

The Federal impulse of 1880 was in the first place a reaction from the ultra-Protectionist policy [of the Victorian colony] of 1878-9 some of whose imposts, and the Stock Tax in particular, being directly aimed at intercolonial imports, naturally provoked great bitterness on the border.

In this, he wasn’t far wrong. After the idea of federating had floated in the ether for years and decades, and an abortive start toward Federation was made in 1890, matters finally came to a head in 1893 when people around the Victoria-New South Wales border gathered in a meeting which later became known as “the Corowa Conference” to push for action on Federation because they were sick and tired of paying customs every time they moved goods across the river.

Back to Deakin’s ‘Federal Story’:

Dread of German aggression in New Guinea and of a French annexation of the New Hebrides [was among] the chief operating causes of [the Intercolonial Convention of 1883].

Deakin himself said:

... that they [the Australasian Colonies] were asked [by the British government] to surrender the New Hebrides as of little commercial value and in the next breath were told that the French set the greatest store by them for commercial development. [The French’s] interest in Australasia were spoken of as large, while ours which were incomparably larger were brushed aside as of no account. [...] We were assured that our alarm as to French intentions was groundless but we should never forget that it was while relying on a similar assurance from the Colonial Office, our trust had been betrayed by a surrender of part of New Guinea to Germany.

So, intercolonial taxes and mutual defence were two of the main issues.

Henry Parkes, the so-called “Father of Federation” repeatedly stressed the issue of mutual defence, starting with his Tenterfield Oration:

The Imperial General who inspected the troops of the colony had recommended that the whole of the forces of Australia should be united into one army. It would be pleasing if they could rely on being safe without taking military precautions at all; but as this was impossible, they must take measure to defend themselves

There was actually strong resistance in Sydney and New South Wales, because of the expected increase in taxation, the imposition of Australia-wide tariffs on imports to the new country (New South Wales had no such tariffs – one reason it resented Victoria’s tariffs on all imported goods, including those from NSW), the unfairness of its representation in the proposed Commonwealth Parliament, and so on. There was especial opposition to Clause 87 of the proposed Consitution, which allowed the future Commonwealth to keep up to 25% of all revenues it raised (the remainder to be distributed back to the states, to offset their revenue losses due to tariffs no longer being collected by them). New South Wales was adamant that this clause, proposed by Tasmanian Premier Edward Braddon and therefore referred to by them as “the Braddon Blot”, was unfair and took too much revenue from their colony. The NSW Premier, George Reid, even threatened to withdraw from the whole process. A sunset clause was inserted, restricting this clause to only ten years’ operation. Instead of federation being driven by a desire to help out other colonies, it nearly didn’t happen because the largest colony didn’t want to subsidise the other colonies.

And, the Western Australian Premier had to be blackmailed by a large chunk of his own citizens who threatened to secede from his colony in order to join this upcoming Federation!

But, the main two reasons for federating were to remove intercolonial taxes and mutual defence.

(Excuse me for re-using a previous answer I've posted on this topic.)