A seemingly relevant entry in Paul Dickson, The Dickson Baseball Dictionary (1989) applies specifically to relief pitchers:
ham-and-egg reliever n. Relief pitcher who is usually brought in after the game has been decided. "He is reliable but nondescript," says Patric Ercolano in his Fungoes, Floaters and Fork Balls , "like a meal of ham and eggs."
I have listened to radio play-by-play of baseball games for decades, in various parts of the country—Houston, Baltimore, New York, and the San Francisco Bay Area—and never heard an announcer use that expression. If the speaker that the OP quotes were using "ham-and-egg" in the sense that Dickson reports, it would probably mean that the game was probably a blow-out, and that the Diamondbacks were either so far ahead or so far behind that they just wanted the less valuable relief pitchers in the bullpen to come in and eat up some innings so that the key relievers would be fresh for the next day's game. The normal term I've heard used in such situations is "mop-up duty" or "mopping up."
But the box score for the game in question—San Diego Padres at [Phoenix] Arizona Diamondbacks, May 10, 2015—indicates that the game was anything but a blowout: the Diamondbacks won 2–1, getting solo home runs in the second and third innings and withstanding a single run by the Padres in the sixth inning. The odd thing about the game wasn't that the Diamondbacks recognized that the game had been decided early and so went to their mop-up men; it's that the starting pitcher lasted only 3⅓ innings despite giving up only 2 hits and 2 walks and throwing only 56 pitches.
But ESPN's recap of the game explains why things unfolded as they did:
PHOENIX -- Daniel Hudson gave the Arizona Diamondbacks all he had.
Making his first major league start since June 2012, the right-hander pitched into the fourth inning Sunday before exiting with a two-run lead. Four relievers held on, finishing a 2-1 victory over the San Diego Padres.
Aaron Hill and A.J. Pollock homered for the Diamondbacks. Hudson, limited to 56 pitches in 3 1/3 scoreless innings, had been used exclusively out of the bullpen since returning late last season from two Tommy John surgeries.
The game was the fourth in four days between the two teams. Arizona had won 11–0 on May 7, a game in which the Diamondback starter had pitched seven of the nine innings—and Hudson had pitched the eighth inning in relief. On May 8, the Padres had won 6–5, with the Diamondbacks' starter going only 4⅔ innings and five relievers seeing action; and on May 9,the Padres had won 6–4 in 12 innings, with the Snakes using six relief pitchers over the final five innings.
So for the fourth game of the series the D-backs had started a guy who was still coming back from Tommy John surgery, had pitched an inning of relief three days earlier, and hadn't started a game in three years. They had used four of the pitchers in their bullpen in consecutive games (both close losses), so those players (including Brad Ziegler, who I think was their closer at that point), were almost certainly unavailable to pitch again. That left the other half of their bullpen—including two pitchers with ERAs over 6.00 and a third with an ERA over 4.00—to see the game through, once the starter faltered.
Under the circumstances, in saying that Arizona was "trying to ham-and-egg it," the announcer probably meant that the Diamondbacks were going to try to grind out a win by turning to a series of undistinguished relief pitchers because that's all the team had available. But if so, he was using "ham-and-egg" in a significantly different sense from the one that Dickson lists in his dictionary.