Both sides are equal since they count the same set: the RHS counts naturals $\rm\:\le n\:x\:$. The LHS counts them in a unique mod $\rm\ n\ $ representation, $\:$ viz. $\rm\ \: j \:\le\: x+k/n\: \iff \ j\:n-k \:\le\: n\:x\:,\ \ j>0 \le k < n\:$.
REMARK $\:$ That every natural has a unique representation of form $\rm \: j\:n-k \ \ \:$ where $\rm\ \ \: j>0 \le k < n\ \ \ $ is simply a slight variant of the Division Algorithm where one utilizes negative (vs. positive) remainders.$\ \ $ To derive this negative form, simply perform the following transformation on the positive remainder form $\rm\ q\: n + r\ \to\ (q+1)\:n + r-n\ $ if $\rm\ r\ne 0\:$, i.e. inc the quotient, dec the remainder by the dividend.
Thus the result is equivalent to the Division Algorithm, whose normal proof is indeed by induction. One could give a direct inductive proof of the result if, instead of invoking the Division Algorithm by name, one unwinds or inlines this inductive proof directly into the proof of the result - much as the same way that the classic Lindenmann - Zermelo direct proof of unique factorization of naturals inlines a division / Euclidean algorithm based descent proof of the fundamental Prime Divisor property $\rm\ p|ab\ \Rightarrow\ p|a\ \ or\ \ p|b\:$.