Welcome to Android Central! Which LG phone? MMS requires mobile data, so if your mobile data was off, or if the mobile data reception is very poor, or if your phone's mobile data settings are configured incorrectly, them MMS messages won't show up correctly. It'll just say "Tap to Download," and if you don't restore your mobile data connection within a certain time, the message will expire and you won't be able to access it.
So, is your mobile data turned on?