There are two discrete steps involved here. I'll list them in reverse order.
2) Morality in Reference to Goals
Morality can be objective when it is relative to some agreed-upon goal. That is, if two or more people agree that a given thing is usually preferable compared to another thing (like eudaimonia vs suffering, life vs death, peace vs war), then, in reference to that goal, there are objectively superior ways to reach those goals compared to other ways. In other words, if two people have the goal of maximizing happiness for themselves and their community, then if you analyze their behaviors and decisions, there are distinctly better ways of behaving compared to other ways. Just like anything else- there are techniques for building a house that result in a better house than other techniques. It doesn't mean it's linear- there can be multiple optimal ways. But there will be some that are better than others.
1) Defining Those Goals
But then, you may point out that there is no necessity to agree on a goal, or that the goals are arbitrary. That's true, but it's not really as chaotic as that. It's no coincidence that most people desire happiness- most of us are hardwired to because that's what is conducive for survival. Life that pursues death or actions that make life unsustainable generally doesn't last very long. For humans, who are social animals who's biggest strengths are intelligence and cooperation, the ability to successfully work with others leads to life or death in the long run. Some goals are inherently more sensible than others for life, because life has gone through an iterative filter to rule out things that don't work. Different things work for different species.
So, morality does have objective aspects for any person any of us would be interested in talking to (basically, non-sociopaths), because we agree on certain rather self-evident goals.