僕 means “I” used by men in semi-formal speech.
は is the particle used to indicate the topic of the sentence.
ここ means “here”.
で結構 following a noun means “fine with something” or “something is good enough”. Also, 結構 often used with the particle は means “I’m fine without it so I don’t need it”.
です is the polite form of だ which is an auxiliary verb for assertive.
So 僕はここで結構です means “I’m fine with here” so it can be translated as “I’ll be fine right here”.
から means “because” which indicates the reason that he said “お気になさらず” meaning “don’t worry about me”. から usually comes in the middle of a sentence so the whole sentence is supposed to be “僕はここで結構ですから、お気になさらず”. But he inverted the clause order for emphasis or dramatic purpose which is a method called “倒置法” (inversion).