What does 这鸡儿啥东西 mean?
I guess the only really confusing part here is 鸡儿, the other parts are quite self-explanatory:

这 = this
啥 = what
东西 = thing

鸡儿 is a curse word from Southwest Mandarin.
Here's an entry from 《四川方言词典》

词语: 鸡儿
发音: ji1 er2
定义: (名) ①雏鸡。②&lt;粗&gt;男阴，骂人语，有的地方只有此义 ▷啊哟!你才香喃?比五香狗肉还香!只默到想巴结你么?哼，一条臭～! (山259)

It's essentially the same as 鸡巴 (cock) and is used almost in the exact same way.
Side-note: although the given pronunciation is ji1 er2 - you're more likely to hear it as a single syllable jier1 and even juer1 in some locales.
So 这鸡儿啥东西 basically means: wtf is this thing?