There are multiple possible motivations, some of which he listed in various letters and speeches. Amongst his most famous was the idea that "A house divided against itself cannot stand". However, there was a lot more to it than that, and one of the most obvious places to look is a particular speech.

In his inaugural address in 1861, Lincoln gives us a glimpse into his belief:

But if destruction of the Union by one or by a part only of the States be lawfully possible, the Union is 'less' perfect than before the Constitution, having lost the vital element of perpetuity.

The idea of perpetuity is very important to Lincoln, throughout his speech. The idea of keeping the Union together because the Union cannot make exceptions, lest it divide itself into even more nations over time over any issue when a minority feels threatened, was one of Lincoln's biggest. This is apparent as he continues:

Plainly the central idea of secession is the essence of anarchy. A majority held in restraint by constitutional checks and limitations, and always changing easily with deliberate changes of popular opinions and sentiments, is the only true sovereign of a free people. Whoever rejects it does of necessity fly to anarchy or to despotism. Unanimity is impossible. The rule of a minority, as a permanent arrangement, is wholly inadmissible; so that, rejecting the majority principle, anarchy or despotism in some form is all that is left.

The idea was rooted in precedent; if this was allowed to occur, then it would set a nasty precedent. Anarchy would reign, he believed, and the Union would fall apart into squabbling minorities. He probably thought similarly that it would leave the federal government powerless as it was under the Articles of Confederation, which didn't really "work" for the majority of people. This is clear when he mentions near the beginning of the speech that the Constitution was made to ordain a "more perfect Union" than that of the Articles. He continues:

Physically speaking, we can not separate. We can not remove our respective sections from each other nor build an impassable wall between them. A husband and wife may be divorced and go out of the presence and beyond the reach of each other, but the different parts of our country can not do this. They can not but remain face to face, and intercourse, either amicable or hostile, must continue between them. Is it possible, then, to make that intercourse more advantageous or more satisfactory 'after' separation than 'before'? Can aliens make treaties easier than friends can make laws? Can treaties be more faithfully enforced between aliens than laws can among friends? Suppose you go to war, you can not fight always; and when, after much loss on both sides and no gain on either, you cease fighting, the identical old questions, as to terms of intercourse, are again upon you.

Here, he points out the difficulties involved. How difficult would it be, for the North and South to suddenly change their entire modes of operation? Taxation would be different, the systems of government were different, and the two would have to negotiate as tense treaty-makers, not as one Union that participated fully in the same government. In short, it would be a lot harder for anything to get done, and the Union would be entirely different for both. Lincoln very clearly didn't like that idea, and so he argued against it.

This should give you a pretty good idea of Lincoln's motivations for keeping the Union together at the outset of the war, and it seems that he didn't really change any of those opinions throughout the war that I saw from my readings of his letters/speeches. Maybe he tweaked how he represented them, but you get the idea.

Source of speech text: http://www.presidency.ucsb.edu/ws/?pid=25818

Follow on question: Although the election of Lincoln is seen as the 'straw that broke the camel's back' and caused the start of Civil War, it seems as though tensions were high on both sides of the politically charged issue of slavery.

With that being said, was it only a matter of time before the Civil War erupted (I understand this question is speculative), and were any of the other candidates of the 1860 election more open to the idea of disbanding the Union?