#R, 159 bytes

f=function(m){M=m%/%10;a=cbind(0,t(apply(M,1,diff)));b=rbind(apply(M,2,diff),0);a[!!a]="|";b[!!b]="_";M[]=gsub("0"," ",paste0(a,b));write(t(M),"",ncol(m),,"")}

With newlines and indentations:

f=function(m){
    M=m%/%10
    a=cbind(0,t(apply(M,1,diff))) #row-wise difference
    b=rbind(apply(M,2,diff),0) #column-wise difference
    a[!!a]="|"
    b[!!b]="_"
    M[]=gsub("0"," ",paste0(a,b)) # M[] is a trick to force the result to have the same structure as M
    write(t(M),"",ncol(m),,"")
    }

Does the integer division of the matrix, measures the row-wise and column-wise differences, and when not null replace by |and _ respectively, then pastes both (painless, thanks to R's vectorization) and outputs.

Test cases:

> m=matrix(c(0,10,20,30,40,50,60,70,80,90,0,0,10,10,20,20,30,30,40,40,0,0,0,10,10,10,20,20,20,30,0,0,0,0,10,10,10,10,20,20,0,0,0,0,0,10,10,10,10,10,0,0,0,0,0,0,10,10,10,10,0,0,0,0,0,0,0,10,10,10,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0),byrow=T,ncol=10)
> f(m)
  |_|_|_|_|_|_|_|_|_
    |_  |_ _|_ _|_ _
      |_    |_ _  |_
        |_      |_ _
          |_        
            |_      
              |_    
                |_  
                  |_
                    
> m=matrix(c(5,5,5,5,5,5,5,5,5,5,5,5,10,10,10,10,10,10,10,10,10,5,5,10,15,15,15,15,15,15,15,10,5,5,10,15,20,20,20,20,20,15,10,5,5,10,15,20,25,25,25,20,15,10,5,5,10,15,20,25,30,25,20,15,10,5,5,10,15,20,25,25,25,20,15,10,5,5,10,15,20,20,20,20,20,15,10,5,5,10,15,15,15,15,15,15,15,10,5,5,10,10,10,10,10,10,10,10,10,5,5,5,5,5,5,5,5,5,5,5,5),byrow=T,ncol=11)
> f(m)
   _ _ _ _ _ _ _ _ _  
  |                 | 
  |    _ _ _ _ _    | 
  |   |         |   | 
  |   |    _    |   | 
  |   |   |_|   |   | 
  |   |         |   | 
  |   |_ _ _ _ _|   | 
  |                 | 
  |_ _ _ _ _ _ _ _ _|