Learn French: Le livre est très amusant.

Because livre (book) is a masculine noun. The rule of adding an e to the end of an adjective only applies when you are describing a feminine noun.

Do you pronounce the t at the end of amusant?

No, the t is silent here. However, if you were describing a feminine noun, you would add an e to make it amusante, and then you would pronounce that t sound!

Is there a connected sound between très and amusant?

Yes! This is called a liaison. Because très ends in a normally silent s and amusant starts with a vowel, the s wakes up and is pronounced like a 'z' sound that links the two words together.

I thought words ending in 'e' were usually feminine. Why is it Le livre?

While many French words ending in e are feminine, there are plenty of exceptions. Livre is one of them, so you just have to memorize that it is masculine.