It is a common structure
[用 (object) 来 (verb)] = [use (object) to (Verb)]
[用 (这种方式) 来 (赞扬京剧演员)] = [use (this method) to (praise) Peking Opera actors]
方式 is the object; 赞扬 is the verb; 京剧演员 is the indirect object
[用(水) 来 (灭)火] = [use (water) to (put out) fire]
水 is the object; 灭 is the verb; 火 is the indirect object
"用这种方式赞扬京剧演员" and "用水灭火" mean the same as "用这种方式来赞扬京剧演员" and "用水来灭火". The difference is adding 来 strengthen the connection between the noun phrase and the verb phrase.
The connection between noun and verb in "use water put out fire" is not as strong as "use water to put out fire"
"用枪杀敌" (use gun kill enemy) might be broken into two unrelated phrases if you say it like "用枪， 杀敌" (use gun, kill enemy). While "用枪来杀敌" (use gun to kill enemy) cannot be broken into two unrelated phrase no matter how you say it