The multibit.key file is just a text file. You should be able to open it with Notepad (or the text editor of your choice) and see your private keys. The keys should start with a 5, K or L and look like a bunch of random characters (a bit like a bitcoin address)... something like this:
5Kb8kLf9zgWQnogidDA76MzPL6TsZZY36hWXMssSzNydYXYB9KF
FYI: DO NOT post these online or send them to anyoneOnce you have your private keys, you don't have to sweep them into Electrum... you can just import the private keys into a new wallet (see steps below). Sweeping is just recommended as it means the coins are placed into a standard HD wallet and can be recovered using just the 12 seed words used to create the HD wallet, rather than having to back up each and every private key for every address you ever use.
I would highly recommend that you do not continue to use MultiBit Classic. It is very old and is no longer maintained or supported by the developers. MultiBit HD has some major issues at the moment, so until the devs fix that, I would not recommend MultiBit HD either.
Also, please note, that importing the keys into Electrum doesn't mean that they disappear from MultiBit Classic. You're really just creating a copy. So if anything goes wrong, you'll still have MultiBit Classic as a backup. As long as you have your private keys (and no-one else does) you'll be able to access your coins.
If you have any other questions, just ask.
To do an import into Electrum:
Step 1. Select "File -> New/Restore"
Step 2. Give your new wallet a name (I suggest something descriptive, so you can tell which wallet is which)
Step 3. Create the wallet file (you can change the storage location on this screen).
Step 4. Select the wallet type, I suggest you leave as "Standard Wallet"
Step 5. Select the type of keystore. Here you MUST select "Use public or private keys" to be able to import.
Step 6. Copy/Paste the private keys from your multibit.key file (NOTE: one key per line... just the key, don't include the datetimestamps at the end!)
Step 7. Create a password. (I also recommend ticking the "Encrypt Wallet File" box for added security)
Step 8. After a (hopefully short) wait... you'll see all the transaction history for all your addresses/keys
Optional: If you want to double check the addresses imported, goto the addresses tab:
If you don't see the "Addresses" tab, you can enable it using: "Wallet -> Addresses":
If you don't see any addresses on the "Addresses" tab, click the little ">" symbol(s) so they become "v" symbols