Walk the words!

Dyalog APL, 63 bytes

s@(n+11 9∘○¨+\0j1*⊃¨,⍨\(8∘≠⍴¨⊢)0,'rdluRDLU'⍳¯1↓s)⍴∘'',⍨2×n←≢s←⍞

uses ⎕IO←0 and features from v16 (@)

n←≢s←⍞ raw input s and its length n

⍴∘'',⍨2×n create a 2n by 2n matrix of spaces

s@(...) amend the matrix with the characters of s at the specified (pairs of) indices

how the indices are computed:

¯1↓s drop the last char of s

'rdluRDLU'⍳' encode 'r' as 0, 'd' as 1, etc; other chars as 8

0, prepend a 0

(8∘≠⍴¨⊢) turn every 8 into an empty list, all others into a 1-element list

,⍨\ cumulative swapped concatenations (abcd -> a ba cba dcba)

⊃¨ first from each

0j1* imaginary constant i to the power of

+\ cumulative sums

11 9∘○¨ real and imaginary part from each; get coords in the range -n...n

n+ centre them on the big matrix

Charcoal, 29 27 20 19 bytes

ＦＳ«Ｆ№rdlu↧ι≔ιω✳∨ωrι

Try it online! Link is to verbose version of code. Explanation:

ＦＳ«

Loop over the input characters.

Ｆ№rdlu↧ι

If the current letter is a direction...

≔ιω

then update the current walking direction.

✳∨ωrι

Print the character in the current walking direction, defaulting to right if no direction has been set yet.

C#, 525 474 Bytes

Edit: Saved 51 Bytes thanks to @steenbergh

It's not pretty, but it does work...

Golfed:

string W(string s){var l=s.Length;var a=new char[2*l+1,2*l+1];int x=2*l/2;int y=2*l/2;int d=0;for(int i=0;i<l;i++){switch(char.ToUpper(s[i])){case'U':d=3;break;case'D':d=1;break;case'L':d=2;break;case'R':d=0;break;}a[y,x]=s[i];switch(d){case 0:x+=1;break;case 1:y+=1;break;case 2:x-=1;break;case 3:y-=1;break;}}string o="";for(int i=0;i<2*l+1;i++){string t="";for(int j=0;j<2*l+1;j++)t+=a[i,j]+"";o+=t==string.Join("",Enumerable.Repeat('\0',2*l+1))?"":(t+"\r\n");}return o;}

Ungolfed:

public string W(string s)
{
  var l = s.Length;
  var a = new char[2 * l + 1, 2 * l + 1];
  int x = 2 * l / 2;
  int y = 2 * l / 2;
  int d = 0;
  for (int i = 0; i < l; i++)
  {
    switch (char.ToUpper(s[i]))
    {
      case 'U':
        d = 3;
        break;
      case 'D':
        d = 1;
        break;
      case 'L':
        d = 2;
        break;
      case 'R':
        d = 0;
        break;
    }
    a[y, x] = s[i];
    switch (d)
    {
      case 0:
        x += 1;
        break;
      case 1:
        y += 1;
        break;
      case 2:
        x -= 1;
        break;
      case 3:
        y -= 1;
        break;
    }
  }
  string o = "";
  for (int i = 0; i < 2 * l + 1; i++)
  {
    string t = "";
    for (int j = 0; j < 2 * l + 1; j++)
      t += a[i, j] + "";
    o += t == string.Join("", Enumerable.Repeat('\0', 2 * l + 1)) ? "" : (t + "\r\n");
  }
  return o;
}

Explanation:

Uses a two-dimensional array and the d value to increment the position of the array in the correction direction, where d values are:

0 => RIGHT
1 => DOWN
2 => LEFT
3 => UP

Test:

var walkTheWords = new WalkTheWords();
Console.WriteLine(walkTheWords.W("codegolf and programming puzzles"));

                            cod                              
                              e                              
                              g                              
                              o                              
                         dna fl    sel                       
                                     z                       
                         p           z                       
                         rogramming pu