Basically Tybalt hates Romeo and calls Romeo a "villain". Romeo says something about love towards Tybalt he says hes not a villain and he tells Tybalt that he doesn't understand him. Romeo says goodbye (he doesn't want to fight).
The bolded part is Tybalt not accepting what Romeo is saying. He says it doesn't "excuses the injuries", which means he doesn't forgive him (doesn't excuse Romeo's previous behaviour). Therefore "Turn and draw" means get ready to physically fight (draw out your sword)
