On the versions of the Play Store with the new "simplified" permissions dialog, you can still find a full list of permissions from the app's store page (the screen with the icon at the top, description, screenshots, &c.). Scroll down to the bottom of the page. Under "Additional Information" you'll see "Permissions" with a link "View details". Click this link to see a dialog with the list of individual permissions.
In particular, an SMS app might have permissions "edit your text messages", "receive text messages", "read your text messages", "send SMS messages", each one saying whether it applies to SMS, MMS, or both. They're pretty self-explanatory on the face of it, but there are some subtleties.
"Receive text messages" means the app will be notified (by Android) of a new message as soon as it arrives. This is normal for an SMS app, or for a few apps that use a special SMS to validate your phone number with some online service (such as Facebook).
In contrast, "read your text messages" lets an app get data from Android's database containing your message history. This means it can see messages sent from different apps, or messages you sent or received before you installed the new app. Any text message you can see in the stock SMS app, this app will also have access to. Again, this is normal in an SMS app.
"Edit your text messages" gives the app write access to that same database. For example, the app uses this to save messages or drafts to your message history. Again, this is completely normal for an SMS app. But it could also be used for an app that illicitly sends premium SMS messages to delete the records afterwards.
Like any permission, it comes down to trust. It's on you to decide if you trust the author of this app to only use these abilities the way you want: not just now, but in future versions of the app too until you you uninstall it.