Q:

trouble understanding an awk scrpt

I'm having having issues understanding this gawk command.  In summary it parses two log (local access) Tomcat log files for certain criteria. 
Here is the function:
gawk 'function getip(rec) {n=split(rec,a,"\"");
        split(a[n-1],ip,",");return ip[1]}
    $10 ~ /302/ && $6 ~ /POST/ && $7 ~ /^\/sso\/lfe\/ui\/login/
    {lfe_user_ip=getip($0);
        user_path[lfe_user_ip]=user_path[lfe_user_ip]"_login-302"}
    /\/sso\/pfe\/rs\/profile\/customer/ && $6 ~ /PUT/
    {pfe_user_ip=getip($0);
        if (user_path[pfe_user_ip] ~ /_login-302/) {
            if ($10 ~ /200/) successful_redirect_conversion+=1;
            else failed_redirect_conversion+=1;} }
    END {print successful_redirect_conversion, failed_redirect_conversion}' \
        localhost_kff_access_log.2017-10-09.txt \
  /ses/var/late/dev001/tomcat/kfc/s002/logs/localhost_kfc_access_log.2017-10-09.txt

What I don't understand is the split method and also the escaping of the path as such:
$7 ~ /^\/sso\/lfe\/ui\/login/

A:

The first split divides the input on literal double quotes and puts the results in a:
n=split(rec,a,"\"");

The value of n indicates how many fields we ended up with.   The next split takes the second to last of these fields and splits that on commas:
split(a[n-1],ip,",");

Finally, the function returns the first of these values:
return ip[1]

Without access to your input logs it's hard to say exactly what this accomplishes, but typically, something like
various values "some of them in quotes"  and "11.22.33.44,55.66.77.88,some more"

will take the last double-quoted field (the n-1 is because split returns an empty field from after the last ") and further remove anything after the first comma, yielding the final result 11.22.33.44. 
Regarding the backslash, you have to escape any literal backslash in Awk regexes because the backslash itself is a regex delimiter. 
$7 ~ /foo/bar/

is a syntax error because the slash looks like the regex ends after foo and the bar/ is outside.   Putting in a backslash like /foo\/bar/ allows the Awk parser to properly see that the internal slash is part of the regex, not its terminating delimiter.