free cash flow

When we use the FCF approach it gives the total value of the company, and to get the value of the equity we subtract the long-term debt only.

To try and explain why, suppose the company was financed entirely from equity. Then using FCF would give us the total value of the equity (we would not then subtract short-term borrowings).
Normally, of course, the long-term finance comes partly from equity and partly from long-term debt. So we do the same thing, but the total value of the company is the total value to the long-term financiers – subtract the long-term debt and we are left with the value of the equity.