Can you mine without transactions? As far as I know, mining means verifying transactions and thus keeping the Bitcoin network alive.
Today, at a local meetup, some people have informed me that mining doesn't need transactions. They said that even if there are no transactions waiting to be verified at all, you can just mine away based on the previous block's "hash", and as soon as you find the right salt you create a new block.
That would mean you don't do anything by "mining" except for blowing a bubble. Where is my logic flawed?
This is essentially correct: the protocol leaves it up to the miner to decide which transactions  to include in a block.  There is no requirement for them to include any transactions at all, other than the "coinbase" transaction which specifies where to send the block reward.

However, most transactions have fees attached, which the miner can collect only by including those transactions in the block.  So this gives the miner an incentive to include transactions if there are any outstanding.

Note that a block with no transactions does still contribute to the security of the currency: it increases the amount of hashing that an attacker would need to do to reverse a transaction recorded in a previous block.  (They would have to produce a chain showing more work than the current one, including the work proved by the zero-transaction block.)

See also Does mining activity only exist if there are bitcoin transactions?.