Let me add some depth to your question:
Why it's recommended to brush your teeth? Put on perfume/cologne, comb your hair or in general be clean — regardless of being Tahir?
Additionally you could ask why do we have to make wudu? First right then left? Why 5 times? Why sit down? Why not pray while I'm jogging?
At the high level, the whole notion of it to admonish your soul and body all together.
Our souls need to be reminded of right and left. Right and left in wudu represent heave and hell. Getting your book of deeds with your right hand vs. left.
Allah wants us to detach from everything and only to face him with our best looks.
The effect of prayer on hijab:
The spirit of prayer is something that is to be within us through out the day. It's not something we do and should then forget completely. When I pray, I should for a good few minutes be reminded of right/wrong, heaven/hell, book of deeds. Allah wants every women to be reminded of the importance of hijab for a few more minutes every day.
Similarly the spirit of prayer is one of the main forces for us to have hijab, cleanliness and tidiness throughout the day. Salat is just is the all-encompassing opportunity to rehearse it all together.
The effect of hijab (and other things) on prayer:
Since having hijab itself is a religious deed and it itself has many facets, doing such during prayer will have a spiritual affect on the praying act itself.
Allah and his angels are present:
Imam Sadiq said:
اعبد الله کأنّک تراه فان کنت لا تراه فانه یراک
Worship Allah as if you're seeing him. If you can't see him then...he can see you
(Usul al-Kafi Vol. 2, Pg. 68)
Having hijab is somewhat like a formal attire.