What is payment batching? How does Bitcoin transaction batching work exactly?  How does this end up looking compared to a regular block of a bunch of transactions?  Also, I have read people claim batching causes weaker privacy, how is this exactly?
How does Bitcoin transaction batching work exactly?

Instead of paying multiple people using a new transaction for each person, you make one transaction that pays those multiple people.
For example, suppose you need to pay 3 people. Instead of making 3 different transactions with one output each, you make one transaction with 3 outputs to the 3 people you are paying.

How does this end up looking compared to a regular block of a bunch of transactions?

It is cheaper partially because each transaction has a fixed amount of overhead. By batching transactions, you only have to pay for the overhead once, not multiple times. Furthermore, usually each transaction will have a change output, and many times that change output must be spent again in the next transaction. This means that if you are sending to 3 people, you are creating 3 change outputs, and 2 inputs that have to spend from a change output. By batching, you only need to make one change output and you don't need to have those 2 additional inputs that spend the change. Those effectively cancel out and are thus removed from the transaction.
In a normal case with paying 3 people:
Input 1 -&gt; Person 1
        -&gt; Change 1 -&gt; Spend Change 1 -&gt; Person 2
                                      -&gt; Change 2 -&gt; Spend Change 2 -&gt; Person 3
                                                                    -&gt; Change 3

By batching:
Input 1 -&gt; Person 1
        -&gt; Person 2
        -&gt; Person 3
        -&gt; Change 3

So you have cut out Change 1 and 2, and Spend Change 1 and 2 which means you are paying less transaction fees.

Also, I have read people claim batching causes weaker privacy, how is this exactly?

All of the people being paid in a batched transaction can see who else you are paying and how much. However it is hard for observers to also know the true identities of those people.