If you look at TweetCaster's permissions on Twitter’s Apps settings screen, it should show permissions to direct messages:
If this does not appear, it's likely something caused the permission to not be granted properly at validation.
Removing your account from TweetCaster (go to the gear icon from the menu, select accounts then tap "Edit” in the upper right and then tap the "-" next to the account name) then re-adding it manually should solve the issue.