I saw some transactions where an input address also appears as an output address. For example, this transaction has one input and three outputs.
Input Address/Amount Output Address/Amount
A / 3BTC B / 0.5BTC
C / 0.5BTC
A / 2BTC
Does this simply mean A paid itself in this transaction? Or does it mean something else?
Yes, in your example the sender assigned some funds to the same output script that they just spent coins from. Address reuse is permitted, but should be avoided as it unnecessarily reveals common authorship of transactions.
Funds are not actually accumulated on addresses but each output is tracked separately in the form of unspent transaction outputs (UTXOs). If you receive two payments that both credit Address A, you will have two UTXOs that have the same spending conditions which still must be consumed in two separate inputs, even if you want to spend them in the same transaction.
When spending a UTXO, it is consumed in full. The available value may then be assigned to outputs. The remainder of funds not assigned to a recipient becomes the transaction fee. If, in your example, Alice had a single UTXO of 3 BTC value and wanted to send 0.5 BTC to Bob and 0.5 BTC to Carol, she would have a remainder of 2 BTC. To prevent paying 2 BTC in transaction fee, she added a so-called change output to reclaim the remaining funds. As described above, it would be preferable, if Alice had used a new address A' that she also controls.
The difference between the total input amounts and the total output amounts goes to miners as a transaction fee. Therefore if you want to send an amount materially lower than an amount you have in one of your UTXOs you need to send some Bitcoin back to yourself. It doesn't need to be the same address as from where you sent from (many would recommend not reusing addresses) but it does need to be to an address that you control (have the private key to).
