While you’ve not really covered trains yet (though you’ve seen the mean←+⌿÷≢ example), sign←÷∘|⍨ can be written as the train (fork) sign←⊢÷| as follows.

Without going into all the details, think of the traditional mathematical notation (TMN) where (f+g)(x)=f(x)+g(x). Forks (3-trains) are just a generalisation of this pattern to all functions (though the middle one has to be dyadic). For n÷|n we have the + become a ÷, while g is |, but what is f‽ We just want n unmodified. This calls for an identity function à la f(x)=x, which is APL’s ⊢. So…
TMN: f(x)=x, g(x)=|x|, sgn(x)=(f÷g)(x)
APL: f←⊢ ⋄ g←| ⋄ sgn←f÷g or simply sgn←⊢÷|

Dyadic + is a scalar function meaning that it will pervasively apply to corresponding elements. Since there are 3 elements on the left, but only 1 on the right (⊂ always gives a scalar result) it will “broadcast” that single element to all of them, so 1 2 3 + ⊂4 5 6 is equivalent to 1 2 3 + (4 5 6)(4 5 6)(4 5 6). Now it pairs up 1 with its4 5 6 and 2 with its4 5 6 and 3 with its 4 5 6, giving (1+4 5 6)(2+4 5 6)(3+4 5 6). Call it a loop if you want .

P.S. Put lines with triple-backticks before and after your code block to make it format in monospace font and respect leading spaces.