Generally speaking, the chemical reaction between the lye and oils (saponification) generates heat and the soap will go through what is called "gel phase" which is what happens when it heats up and turns into a vasaline like texture. People insulate so that the gel reaches all the way to the corner.
Millersoaps.com has a lot of great information you should check out. I also really like the book, Soap Makers Companion.
ETA: if you are using the milky way molds, they can be a PITA to get out sometimes. I second sicking it the freezer. Let it freeze all the way, then take it out and let it start sweating and it should pop out.
And congrats on your first batch of soap!!! It is so exciting isn't it! Where did you order you fo samples from? I'm nosey, lol. I get my dragon's blood from Southern Garden Scents.