Lincoln worked his hardest to keep those states in the Union. A combination of his efforts and a general feeling that the South was a losing bet kept the border states loyal.

• Delaware had no geographical connection to the Confederacy. The state would have had zero chance of successfully defending itself if it had seceded.
• Kentucky declared neutrality. Before the Union river fleet was fully constructed, Lincoln instructed his captains to sail the Ohio River on whatever big boats they could find. It was an effective show of force. And then the Confederates were the first ones to move into Kentucky and violate its neutrality, pushing public opinion towards Union.
• Maryland strongly wanted to secede. Here Lincoln imposed martial law and shut down the press, firmly quelling dissent.
• Missouri went back and forth, but in the end the Union Army had the firmest control over the state, so Missouri never seceded.

It's worth noting that Kentucky and Missouri both sort of seceded--in that Confederate splinters groups held conventions, passed secession resolutions, and even appointed secessionist governments. However, the military reality was that the Union could run roughshod over both states and that the Confederacy could not adequately defend or sustain the secessionist factions.

I would suggest reading Bill Freehling's newish book on the succession crises. One thing that becomes immediatly clear is that there were really 3-5 different souths: border south, middle states (those who left only after lincoln called for troops. These were similar to the border south albeit with a stronger slave base.), the deep south and the deep south variants of Louisiana and South Carolina whose political cultures were markedly different (SC extremely aristocratic and LA with its french catholics, entrepreneurial spirit and less strict color line).

The border south saw a much stronger opposition to the "slaveocracy" especially because as Freehling points out the strong defense of slavery was necessarily against the freedom of the white male population. While most would defend slavery if attacked by outsiders, the problems of free labor (in the "free labor free soil free men" sense) created real intra state conflicts and made opposition to slavery somewhat acceptable especially when combined with colonization (indeed i've come to the opinion that the real source of the civil war and the continuation of slavery was the lack of any practical way for the upper and middle south to both abolish slavery and kick the slaves out of the slave states) and the fact that border states, being close to free territory, suffered a constant leakage of runaway slaves.

So essentially a "solid south" never really existed and when you consider that the middle south didn't join the deep south (which itself sort of got caught up in the moment and easily could have not voted for succession) until after war was inevitable it's clear that one reason the border states didn't join was because they weren't ready to fight to the death to preserve a slave aristocracy which many saw as against their own interests and, of course, to commit treason.

so as freehling points out, Yancy and the fireeaters pushed the issue of succession because they rightly foresaw that over time the border south would continue to betray their interests further weakening the ability of the south to maintain a slave society and thus the CSA tried to force the issue and it was successful enough to force the middle south to join them but the border south was too far gone and didn't rise together with the deep south (though some of the states, especially Missouri, really split down the middle creating a competing civil war governments for the state).

the main source is Freehling's Road to Disunion

here is Foner reviewing the book

http://www.nytimes.com/2007/04/08/books/review/Foner.t.html?pagewanted=all

Freehling examines how, in the 1850s, the Lower South became more and more committed to defending slavery as a positive good. At the same time, he tells us, the Middle and Border Souths persisted in the dream of eventually removing all slaves (and free blacks) from the region and reorienting its economy toward the North. Secessionists were not wrong, he writes, to see these states as slavery’s Achilles’ heel.