Q:

Why do passive ETFs require so much trading (and incur costs)?

As far as I've understood, the idea of passive ETFs is to track an index. Typically, the index is a value-weighted stock market index, such as the S&P500. 
Now, if I'm not mistaken, tracking a value-weighted index is extremely easy - just buy the shares in the exact amount they are in the index and wait. The only time you would need to rebalance your holdings is when there is a change in the index, i.e. a company is dropped and a new one is added, right?
Now, isn't it the case that most stock market indices are revised only a few times per year? If so, why do passive ETFs require frequent rebalancing and generally lose to their benchmark index?

A:

Now, if I'm not mistaken, tracking a value-weighted index is extremely easy - just buy the shares in the exact amount they are in the index and wait.

Yes in theory. In practise this is difficult. Most funds that track S&P do it on sample basis. This is to maintain the fund size. Although I don't have / know the exact number ... if one wants to replicate the 500 stocks in the same %, one would need close to billion in fund size.
As funds are not this large, there are various strategies adopted, including sampling of companies [i.e. don't buy all]; select a set of companies that mimic the S&P behaviour, etc.
All these strategies result in tracking errors. There are algorithms to reduce this.

The only time you would need to rebalance your holdings is when there is a change in the index, i.e. a company is dropped and a new one is added, right?

So essentially rebalance is done to;

Mitigate Tracking Errors [Almost daily]
Change in free float shares, results in change in weight [At times more frequent than quarterly]
Company dropped and added [generally quarterly]

If so, why do passive ETFs require frequent rebalancing and generally lose to their benchmark index?

lets take an Index with just 3 companies, with below price.
The total Market cap is 1000

The Minimum required to mimic this index is 200 or Multiples of 200. If so you are fine.  

More Often, funds can't be this large. For example approx 100 funds track the S&P Index. Together they hold around 8-10% of Market Cap. Few large funds like Vangaurd, etc may hold around 2%. But most of the 100+ S&P funds hold something in 0.1 to 0.5 range.
So lets say a fund only has 100. To maintain same proportion it has to buy shares in fraction. But it can only buy shares in whole numbers. This would then force the fund manager to allocate out of proportion, some may remain cash, etc. As you can see below illustrative, there is a tracking error. The fund is not truly able to mimic the index.

Now lets say after 1st April, the share price moved, now this would mean more tracking error if no action is taken [block 2] ... and less tracking error if one share of company B is sold and one share of company C is purchased.

Again the above is a very simplified view. Tracking error computation is involved mathematics. 
Now that we have the basic concepts, more often funds tracking S&P

don't buy all stocks.
don't buy in same proportion.

Thus they need to rebalance.

