Learn Dutch: Ik reken erop dat je de baby meeneemt. - I count on you bringing the baby.

Erop means "on it". The verb rekenen op means "to count on". In English, you might say "I count on you bringing...", but in Dutch, you can't place a preposition (op) directly before a dat (that) clause. Instead, you use erop to say "I count on it, that you...".

Can I just say "Ik reken op dat je..." instead of "Ik reken erop dat..."?

No, that is incorrect in Dutch. You cannot follow a preposition like op directly with a dat-clause. You must use the combination of er + preposition (like erop) as a bridge to introduce the rest of the sentence.

Why is meeneemt at the very end of the sentence?

The word dat (that) introduces a subordinate clause. In Dutch subordinate clauses, the conjugated verb gets pushed all the way to the end of the sentence.

Meenemen (to bring along) is a separable verb. Why isn't it split up here?

In a normal main clause, it would split (for example: Je neemt de baby mee). However, because dat pushes the verb to the end of the clause, the two parts of the separable verb (mee and neemt) are forced next to each other at the end of the sentence. When they meet there, they glue back together into one word.