There isn't a sole reason for why these states decided to stay loyal to the Union despite being slave states. Each of the border states had their own reasons for not seceding from the Union.

Let's start start off with Delaware since its pretty straight forward and easy to understand. By the mid 1800s, slavery was pretty well confined to the extreme southern portions of Delaware. The rest of the state identified much more closely with the northern states. Also, when you look at Delaware's geographical location, it's pretty clear that secession would not have gone very well for them. Delaware bordered two free states (Pennsylvania and New Jersey) and a slave state (Maryland) that decided not to join the Confederacy. They would have been completely cut off from the rest of the Confederacy, and would have been overrun in no time.

From Delaware we move west where we'll find Maryland. Like Delaware, Maryland's decision to remain loyal to the union was partially influenced by geography and the fact that they shared a long border with Pennsylvania which they would have never been able to defend. This is where the similarities between the two states begin to fade. Somewhat similar to Delaware, Maryland was home to both Northern and Southern cultures and ideologies. The difference was, most of Maryland still relied heavily on agriculture and the use of slaves. Because of this, Maryland was home to a sizable group of people who supported secession and the Confederate cause. Still though, many people in Maryland felt that secession wasn't the answer and that state should stay in the Union. The divide amongst Maryland residents was especially troubling to people in Washington D.C. because, if Maryland seceded, the capital of the US would be behind enemy lines. After growing unrest and a deadly riot in Baltimore, martial law was declared in Maryland. Confederate supporters in the Baltimore city government and Maryland state government were arrested and kept at Fort McHenry. This pretty well ended any chance of Maryland seceding. Most of Maryland's Confederate supporters left the state and went to Virginia to join the war on the side of the South.

Next we have Kentucky. Like the rest if the border states, Kentucky's economy was gradually becoming less dependent on slavery. By the time of the Civil War, many of Kentucky's farmers had switched from growing tobacco in favor of crops like corn which required less labor and skill. This led to many of Kentucky's slaves being sold to the larger cotton plantations in states further south. The decline in the use of slave labor in Kentucky was a big reason for their decision to stay in the Union. While many of Kentucky's government officials supported the Confederacy and it's cause, they didn't feel that secession was the best course of action. That line of reasoning was pretty standard throughout the border states. What set Kentucky apart though was their petition to remain neutral during the conflict. They wanted to remain loyal and economically were not as reliant on slave labor as the other southern states, but refused to send the Union army troops "for the wicked purpose of subduing her sister Southern states." Part of the neutrality agreement stated that neither side was to send soldiers into Kentucky. This part of the agreement was broken when Confederate General Leonids Polk took control of Columbus, Kentucky. Shortly after, General Grant stationed his troops in Paducah, Kentucky. The Kentucky legislature met because of the breech of the agreement, and decided that the Confederacy was responsible for the violation of the neutrality agreement, and exiled only the Confederates from the state. After this, there was a noticeable shift in support towards the Union. There were still a large number of people who supported the South, but any chance of Kentucky officially seceding had pretty much gone out the window. A separate Confederate government did eventually form in Kentucky and was officially admitted into the confederacy as the 13th state. Of course, the US government didn't consider the Confederate government in Kentucky to be legitimate so officially, Kentucky never seceded.

Now we can take a look at Missouri. Missouri's top government officials were strong supporters of the South, and Governor Jackson pushed the legislature to take a vote on secession. The legislature decided to stay in the Union which upset the governor. He decided to take matters into his own hands. Jackson decided to attack the federally controlled St. Louis arsenal and then take over the city of St. Louis, which would give him and the separatists control of the state and would allow them to secede. He asked Jefferson Davis for weapons, which he got, and then set up "Camp Jackson" just outside of St. Louis. The St. Louis arsenal was under the command of Captain Lyons at the time. Lyons suspected that Jackson was going to attack the arsenal, so he went under cover to Camp Jackson dressed as an old woman and was able to figure out Jackson's plan. Lyons and his men took over Camp Jackson and forced the secessionists to surrender. Jackson and his men were arrested and forced to march to the arsenal where they were quickly released. That story is important to Missouri's status as a border state because the forced public march was seen as being intentionally humiliating by secessionists in St. Louis, and caused a good bit of unrest. This escalated into a violent riot that killed about 30 people. After that, martial law was implemented. The whole incident caused a serious divide in Missouri's population. Before that, most of the state wanted to stay neutral like Kentucky, but after that everyone was ready to fight. After the incident, Jackson proposed a "Military Bill" which passed, creating the State Guard which was under the command of Governor Jackson. The bill stated that all adult men in Missouri were reserves in the State Guard, meaning they were all under the direct control of Jackson. This meant that Jackson had become the equivalent of the dictator of Missouri. After that law was passed, Lyons declared war on Jackson and the State Guard. Lyons invaded Jefferson City and ran Jackson and his men out of town. Eventually they fled all the way into Arkansas where they stayed for the rest of the war. A new Unionist governor was appointed and Missouri never officially seceded. The Confederate government led by Governor Jackson was recognized by the CSA and was admitted the the CSA as the 12th state.

Each of the states had their own reasons to remain loyal to the Union instead of joining the confederacy, but there is a similarity between them. They had become less reliant on slaves over time because the crops that they grew didn't require the massive work force that had become necessary on the big cotton plantations of the Deep South. Also, large portions of most of the border states were mountainous, and had no need for large plantations with big slave populations. Slavery was the root cause of secession and the war, and plain and simple, slavery in the border states wasn't as important economically by that time as it was further south.

I will only speak for Maryland. Maryland was fairly well split between northern and southern sympathies, With Washington sandwiched between Maryland and Virginia, Lincoln really gave Maryland no choice. He incarcerated some of the state politicians (including the mayor of Baltimore) who were southern sympathizers and placed troops in Baltimore, Annapolis and along all major railroad and shipping routes. Maryland Governor Hicks moved the state legislature from pro-south Annapolis to pro-union Frederick, Maryland where they voted not to seceed. Lincoln did not abolish slavery in the state however, even after the Emancipation Proclamation. When Lee made his first offensive push into Maryland (ending at Antietam) he hoped the Maryland citizens would rally to his cause, but few did.