How can I optimize this PHP script?

There have been some good answers here, and I'll start by seconding what they've said in a brief summary and then add in my two cents.

(1) Your biggest performance gain will come from Erik's two suggestions regarding cURL. Switching to cURL will provide a small increase in performance (maybe 0.5 to 1 second or more per call), but using multi-curl to call both URLs in parallel will likely provide THE ABSOLUTE BIGGEST BENEFIT of all the suggestions here, no questions asked (since you're doing these network fetches in a loop). Here's a class that someone else wrote that simplifies multi-curl a bit:

<?php
// LICENSE: PUBLIC DOMAIN
// The author disclaims copyright to this source code.
// AUTHOR: Shailesh N. Humbad
// SOURCE: https://www.somacon.com/p539.php
// DATE: 6/4/2008

// index.php
// Run the parallel get and print the total time
$s = microtime(true);
// Define the URLs
$urls = array(
  "http://localhost/r.php?echo=request1",
  "http://localhost/r.php?echo=request2",
  "http://localhost/r.php?echo=request3"
);
$pg = new ParallelGet($urls);
print "<br />total time: ".round(microtime(true) - $s, 4)." seconds";

// Class to run parallel GET requests and return the transfer
class ParallelGet
{
  function __construct($urls)
  {
    // Create get requests for each URL
    $mh = curl_multi_init();
    foreach($urls as $i => $url)
    {
      $ch[$i] = curl_init($url);
      curl_setopt($ch[$i], CURLOPT_RETURNTRANSFER, 1);
      curl_multi_add_handle($mh, $ch[$i]);
    }

    // Start performing the request
    do {
        $execReturnValue = curl_multi_exec($mh, $runningHandles);
    } while ($execReturnValue == CURLM_CALL_MULTI_PERFORM);
    // Loop and continue processing the request
    while ($runningHandles && $execReturnValue == CURLM_OK) {
      // Wait forever for network
      $numberReady = curl_multi_select($mh);
      if ($numberReady != -1) {
        // Pull in any new data, or at least handle timeouts
        do {
          $execReturnValue = curl_multi_exec($mh, $runningHandles);
        } while ($execReturnValue == CURLM_CALL_MULTI_PERFORM);
      }
    }

    // Check for any errors
    if ($execReturnValue != CURLM_OK) {
      trigger_error("Curl multi read error $execReturnValue\n", E_USER_WARNING);
    }

    // Extract the content
    foreach($urls as $i => $url)
    {
      // Check for errors
      $curlError = curl_error($ch[$i]);
      if($curlError == "") {
        $res[$i] = curl_multi_getcontent($ch[$i]);
      } else {
        print "Curl error on handle $i: $curlError\n";
      }
      // Remove and close the handle
      curl_multi_remove_handle($mh, $ch[$i]);
      curl_close($ch[$i]);
    }
    // Clean up the curl_multi handle
    curl_multi_close($mh);

    // Print the response data
    print_r($res);
  }

}

The catch here is that this approach depends heavily on how many trade offers you have at any given time, since you're doing a network call for each one. If you have 1,000 trade offers, you might have to break them up into smaller chunks so you're not slamming the steam API with a ton of calls all at the same time.

(2) If you're running it every 15 seconds, then you're likely incurring some overhead from the script starting up. You could run this script in an infinite loop to eliminate that startup time, although you'd have to ensure there's no memory leaks so your script doesn't eventually run out of memory:

<?php
set_time_limit(0);
while(true)
{
  ...your code here...

  // Wait 15 seconds before the next round
  sleep(15);
}

(3) I'm assuming your database is pretty small, but if you've got 10k records or more in any given table, then indexes ARE going to be important, as Herco mentioned. Without good indexes, your SQL queries are going to suffer.

However, I would focus less on #3 and more on #1 and #2 for your best improvements.