When clover is mowed and the clippings mulched back into the soil, the decomposition of the leaves adds nitrogen to the soil. If you remove the clippings the nitrogen goes with it.

Yes, "green manure" is taking nitrogen fixing crops (like clover and beans and peanuts) and to mulch them while still green, and incorporate that decomposing mulch into the soil you're using. That adds nitrogen in fewer steps than the traditional way of using animal manure (where the nitrogen still ultimately comes from plants).

Of course, the modern Haber process also fixes nitrogen through industrial chemistry rather than agriculture, so most commercial fertilizer today gets its nitrogen from chemical synthesis of atmospheric nitrogen into ammonia.