Here it is spread out:

while (<>) {
    s+${SOURCE}/${SAMS}/++g;
    s+${SOURCE}/${SAMLIB}/++g;
    s+${SOURCE}/${SMCLIB}/+${SOURCE}/${KENO}/+g;
    if ( not m+^${SOURCE}/+ ) {
         s+${SOURCE}/+\$$\{OBJECT\}/+g;
         print
    }
}
[download]

This is run on a file named ".depend", and then something else runs after that.

I'm guessing that ${SOURCE} and friends are being interpolated somehow (even though they're in single quotes) because there's nothing in the Perl to give those variables values.

Anyway, this is using "+" instead of "/" for the delimiters on s///, probably so that it can put a literal slash in there without having to backslash it. What it does (for each line) is something like:

1. Remove "${SOURCE}/${SAMS}/" and "${SOURCE}/${SAMLIB}/" anywhere in the line.
2. Replace every "${SOURCE}/${SMCLIB}/" with "${SOURCE}/${KENO}/"
3. If the line starts with "${SOURCE}/", replace every "${SOURCE}" with a literal "${OBJECT}" (I think), and print the line.
4. (Lines that don't start with "${SOURCE}/" are not printed and so removed.)

I'm guessing that ${SOURCE} and friends are being interpolated somehow (even though they're in single quotes)

Those quotes are processed by the shell. In the bourne shell or a derivative like bash, single quote don't perform any interpolation.

$ echo "${PATH}"
.:/home/ikegami/bin:/usr/local/bin:/usr/bin:/bin:/usr/bin/X11:/usr/gam
+es
$ echo '${PATH}'
${PATH}
[download]

( Here on end, assuming the bourne shell or a derivative like bash )

So Perl sees

BEGIN { $^I = ''; }
while (<>) {
    s+${SOURCE}/${SAMS}/++g;
    s+${SOURCE}/${SAMLIB}/++g;
    s+${SOURCE}/${SMCLIB}/+${SOURCE}/${KENO}/+g;
    if ( not m+^${SOURCE}/+ ) {
         s+${SOURCE}/+\$$\{OBJECT\}/+g;
         print
    }
}
[download]

On the other hand, Perl regexp and replace expressions *do* interpolate. So the above is equivalent to

BEGIN { $^I = ''; }
while (<>) {
    s+//++g;
    s+//++g;
    s+//+//+g;
    if ( not m+^/+ ) {
         s+/+\${OBJECT}/+g;
         print
    }
}
[download]

Depending on what the OP wants to do, he needs to use -e"" instead of -e'', to escape the dollar signs, or to define those variables in Perl.

My guess is that the code snippet is from a make file (while we are guessing).

Ordinarily, I'd agree with you, but I get the impression that the OP's code works as-is, and he's just trying to figure out what it does and how. My guess is that the whole thing is in double quotes, and what perl actually sees is:

BEGIN { $^I = ""; }
LINE: while (defined($_ = <ARGV>)) {
    s[value-of-SOURCE/value-of-SAMS/][]g;
    s[value-of-SOURCE/value-of-SAMLIB/][]g;
    s[value-of-SOURCE/value-of-SMCLIB/][value-of-SOURCE/value-of-KENO/
+]g;
    if (not m[^value-of-SOURCE/]) {
        s[value-of-SOURCE/][$${OBJECT}/]g;
        print $_;
    }
}
[download]

That $${OBJECT} in there still seems wrong, though, and the OP is ambiguous about whether this is working code or not, so I'm pretty far from sure about this.

s[$SOURCE/$SAMS/][]g;
s[$SOURCE/$SAMLIB/][]g;
s[$SOURCE/$SMCLIB/][$SOURCE/$KENO/]g;
if (not m[^$SOURCE/]) {
    s[$SOURCE/][$$OBJECT/]g;
    print $_;
}
[download]

Does that help?