Casting a symbolic vote means casting a vote on a topic so as to take a public stance on it without needing to face any consequences (or because you'll actually face consequences in your constituency if you vote along with the party line). Typically, the bill is certain to get killed down the road, or certain to pass anyway. It's a vote that doesn't matter except in the public's opinion.
For instance, Republicans sent a few bills to Obama's White House that included language to repeal the Affordable Care Act, with the full confidence that Obama would veto them. It did not matter that these bills would get vetoed. What counted was making the public statement that they wanted to repeal Obamacare.
Another example would be when elected officials vote against the party line for the sake of publicly standing when a bill is unpopular (conversely, popular) in their constituency, all while knowing that the majority is comfortable enough that a few dissenting voices (or a few more votes) won't matter. I can't think of a clearcut example off the top of my head, but it happens regularly in Europe for topics that target groups that are concentrated in only a few constituencies (e.g. tax increases, NIMBY topics like authorizing a new airport, or extra layers of government regulation). Think voting against a public health bill that hurts tobacco interests by senators in tobacco producing states.
The key point to understand here is that, more often than not, a bill's outcome is mostly understood before actual voting takes place. Intensive negotiations occur between the legislative and executive branches of government, as well as between the various parties (which is to say two in the US or the UK, but countries in Continental Europe typically have more than that), and between members of parliament within those parties. There also is a lot of "I'll scratch your back if you scratch mine."