I got following message: "After Effects: this project must be converted from version 10.5 (Windows 64). The original file will be unchanged." And then i got this: "Unable to connect to Typekit. Missing fonts cannot be resolved at this time."
The first message is just letting you know that the file you are opening was created with an older version of After Effects. To prevent you from writing over it (for compatibility reasons), After Effects is going to make you save any changes to a new file.
This is not a problem at all. It's just letting you know that you should save it with a new name.
The second message indicates that you're not connected to Creative Cloud so it can't use TypeKit to resolve font differences. It is likely that the project you are opening uses fonts that you do not have currently installed. If you are connected to your Creative Cloud account, TypeKit can find replacement fonts for you so things look the exact same. Otherwise, you can just pick whatever fonts you want for any text layers.
Again, this isn't a big problem. You should still be able to open and work with the project. The only downside is that the fonts will be different.