One of your SHELL dot files has an exit command in it which kills the current Terminal process. Open a Finder Window, and press shift+command+period. This will show your UNIX hidden files in your home directory and depending on the default SHELL these might be:
Bash
- .bash_login
- .bashrc
- .bash_profile
- .profile
Zsh
Click once on the dimmed, individual dot file and press the space bar. The Finder's Quick Look feature will show the contents of each of these text files. When you find the one that has the exit command in it, right-click on it and open it in ideally a programmer's editor of your choice, or TextEdit — and remove the line with the exit command in it. Save the file as plain text. That should resolve the Terminal quitting upon launch.
To restore your Finder Window to its normal view, press shift+command+period again.