Why do people apparently "look up Bitcoin addresses on websites"? I often hear things such as:

Remember, if you look up an address on Blockchain.info or something, they will have a record of your IP address being interested in that address and thus associate it with you.

Sure. I follow that logic. But what I don't understand is why they would do that in the first place. Why use some &quot;website&quot; when the whole point is that Bitcoin is decentralized and you have your own wallet/client/node that you can look at?
And what is their &quot;objective&quot; with inspecting some address that they have just used? Why not look in Bitcoin Core (or whatever they use) to simply see if it's in memory pool or has confirmations or whatever it is they are trying to look up?
@Prince M's answer is great, I have just a couple things to add:
Wallet software will generally only show you transactions relevant to the wallet, so if you ever want to explore transactions beyond your own, you might want to use a block explorer to do so.
If you are running a full node, then you can look up transactions using bitcoin-cli, but a full node does not index the blockchain according to addresses, and so the intuitive way to explore transactions that most block explorer services provide does not really exist when using software like bitcoin-core. Likewise 'address balances' are not explicitly retrievable from a full node, though there do exist some open-source self-hosted blockchain explorers that you can run on top of a node, that will compute and display such things for you. But in any case, simply running a full node does not lay open the blockchain for you in a human-digestible format, getting to that point requires additional work and software.
On another note, some wallets don't have very informative user interfaces. For example, I've seen wallets which do not display the fee-rate your transaction includes, and so if you're waiting for confirmation, just looking at your wallet will not provide any information about where in the mempool your transaction resides. Looking up the transaction using a block explorer may be helpful to the user in this case.
Of course, this can be tragic, as using such an online block explorer service can be detrimental to your privacy. For this reason, I would recommend using a block explorer that offers a .onion domain, so that you can view it over the tor network, to help preserve your privacy.