There are two issues here.
First, you will not normally encounter anyone who checks your immigration status during domestic travel in the US. TSA security screening just checks that you are not bringing prohibited items and checks your ID to make sure you are the person whose name is on the boarding pass. Any foreign passport is an accepted ID for TSA identification purposes. Whether it has a visa or not is irrelevant. TSA does not have the jurisdiction nor is trained to check anything relating to immigration status.
Second, although unlikely, there is a small chance of encountering a CBP checkpoint while traveling within the US, on highways or airports within 100 miles of the border. (It is also possible that you might encounter a rogue TSA agent who thinks it's their job to check visas and doesn't understand that US visas are only for entry, who might call over a CBP officer to check you.) In this case, you would present evidence that you are staying legally in the US.
Note that US visas are sole for entry to the US; the visa's expiration date just means the last date on which you can use that visa to enter. It is perfectly normal to be in status in the US months or even years after the expiration of one's visa, depending on the status. What determines how long you can stay in the US is your I-94, and by other regulations regarding your status. I-94s are electronic nowadays; the date on the I-94 you received at entry is usually also written underneath your entry stamp in your passport. If the date was not written on your passport, or you lost your passport, you can print out your I-94 from the I-94 website. If you received your most recent I-94 from a Change of Status or Extension of Status in the US, the approval for that would have come with a paper I-94. In the unlikely event you are checked for your immigration status, showing your I-94 should suffice.
If "applied for a green card" means that you have filed I-485 for Adjustment of Status in the US, you are allowed to stay in the US even if you are out of status. In that case, you should bring your I-485 receipt with you and show that in the unlikely event you are checked for your immigration status.