The -i flag of sed means to perform the command "in place", that is, save any changes in the input file. Use this flag with extreme caution, one wrong move and you can completely break the original file.
The regex pattern /^Exec=[^ ]*/ will match the line starting with Exec= followed by zero or more non-space characters.
In the replacement string, & -s, the & is replaced with whatever was matched, in this example probably something like Exec=gimp-2.8, after which we add a space and the -s flag which will effectively disable the splash screen when starting Gimp.
The -i flag of sed works differently in GNU and BSD systems. This example works in GNU systems only. The equivalent in BSD is:
sudo sed -i '' 's/^Exec=[^ ]*/& -s/' /usr/share/applications/gimp.desktop
In any case, always be very careful when using the -i flag of sed.
sed is designed for editing streams - editing files is what ed is for! You can get consistent behavior on any UNIX platform with the above one-liner.
The printf command sends a series of editing commands to ed, each separated by a newline. In this case, the substitution command ,s/^Exec=[^ ]*/& -s/ is nearly the same as in sed, appending a space and a -s to the line starting with Exec=. The only difference is the comma at the beginning designating the lines to operate on. This is shorthand for 1,$, which tells ed to apply the command to the first through the last lines (i.e., the entire file). w tells ed to write the file, and q to quit.