Exploded Suffixes

Jelly, 5 bytes

ṫJxJY

Try it online! or verify all test cases.

How it works

ṫJxJY  Main link. Argument: s (string)

 J     Indices; yield I := [1, ..., len(s)].
ṫ      Tail; get the suffixes of s starting at indices [1, ..., len(s)].
   J   Indices; yield I again.
  x    Repeat. The atom 'x' vectorizes at depth 1 (1D arrays of numbers/characters)
       in its arguments. This way, each suffix t gets repeated I times, meaning
       that the first character of t is repeated once, the second twice, etc.
       If left and right argument have different lengths, the longer one is
       truncated, so I can safely be applied to all suffixes.
    Y  Join, separating by linefeeds.

J, 22 12 8 bytes

Thanks to miles for saving 14 bytes!

(#~#\)\.

Now this is a really nice solution. Pretty succinct, too.

This is the hook #~#\ applied to the suffixes (\.) of the input. The hook, when called on input y, is decomposed as thus:

(#~#\) y
y #~ #\ y

Here are some intermediate results:

   ]s =: 's p a c e'
s p a c e
   #\ s
1 2 3 4 5 6 7 8 9
   (quote) s
's p a c e'
   (quote;#)\ s
+-----------+-+
|'s'        |1|
+-----------+-+
|'s '       |2|
+-----------+-+
|'s p'      |3|
+-----------+-+
|'s p '     |4|
+-----------+-+
|'s p a'    |5|
+-----------+-+
|'s p a '   |6|
+-----------+-+
|'s p a c'  |7|
+-----------+-+
|'s p a c ' |8|
+-----------+-+
|'s p a c e'|9|
+-----------+-+
   1 2 3 # '123'
122333
   3 3 3 # '123'
111222333
   ]\. s
s p a c e
 p a c e
p a c e
 a c e
a c e
 c e
c e
 e
e
   quote\. s
's p a c e'
' p a c e'
'p a c e'
' a c e'
'a c e'
' c e'
'c e'
' e'
'e'
   (#~#\) s
s  ppp    aaaaa      ccccccc        eeeeeeeee
   (#~#\)\. s
s  ppp    aaaaa      ccccccc        eeeeeeeee
 pp   aaaa     cccccc       eeeeeeee
p  aaa    ccccc      eeeeeee
 aa   cccc     eeeeee
a  ccc    eeeee
 cc   eeee
c  eee
 ee
e

Test cases

   f =: (#~#\)\.
   f
(#~ #\)\.
   f ''
   f 'a'
a
   f 'bc'
bcc
c
   f 'xyz'
xyyzzz
yzz
z
   f 'code-golf'
coodddeeee-----ggggggooooooollllllllfffffffff
oddeee----gggggoooooolllllllffffffff
dee---ggggooooollllllfffffff
e--gggoooolllllffffff
-ggooollllfffff
goolllffff
ollfff
lff
f
   f 's p a c e'
s  ppp    aaaaa      ccccccc        eeeeeeeee
 pp   aaaa     cccccc       eeeeeeee
p  aaa    ccccc      eeeeeee
 aa   cccc     eeeeee
a  ccc    eeeee
 cc   eeee
c  eee
 ee
e

   ]tc =: <;._1 '|' , '|a|bc|xyz|code-golf|s p a c e'
++-+--+---+---------+---------+
||a|bc|xyz|code-golf|s p a c e|
++-+--+---+---------+---------+
   ,. f &. > tc
+---------------------------------------------+
+---------------------------------------------+
|a                                            |
+---------------------------------------------+
|bcc                                          |
|c                                            |
+---------------------------------------------+
|xyyzzz                                       |
|yzz                                          |
|z                                            |
+---------------------------------------------+
|coodddeeee-----ggggggooooooollllllllfffffffff|
|oddeee----gggggoooooolllllllffffffff         |
|dee---ggggooooollllllfffffff                 |
|e--gggoooolllllffffff                        |
|-ggooollllfffff                              |
|goolllffff                                   |
|ollfff                                       |
|lff                                          |
|f                                            |
+---------------------------------------------+
|s  ppp    aaaaa      ccccccc        eeeeeeeee|
| pp   aaaa     cccccc       eeeeeeee         |
|p  aaa    ccccc      eeeeeee                 |
| aa   cccc     eeeeee                        |
|a  ccc    eeeee                              |
| cc   eeee                                   |
|c  eee                                       |
| ee                                          |
|e                                            |
+---------------------------------------------+

Python, 61 bytes

f=lambda s,i=0:s[i:]and-~i*s[i]+f(s,i+1)or s and'\n'+f(s[1:])

Alternative 63:

f=lambda s,b=1:s and f(s[:-1],0)+s[-1]*len(s)+b*('\n'+f(s[1:]))