The way I see it is that the trans person has had their whole life to understand how they feel and come to terms with being trans, even if it wasn't all conscious. Although it's hard to admit to yourself, and some people spend decades in denial and repression of their true selves, it's possible for us to look back and make sense of it all, because we lived it and felt those feelings. Some family and friends never pick up on the same things as us, or just think we're masculine/feminine for our sex. To the average cis person the last thing that is going to be on their mind is that the person is actually trans but in the closet. And then one day they're told that the person they care about is actually the other gender to what they spent a long time believing. They have none of the background to make sense of it all with, it's not a gradual process, it's sudden.
At the point we come out to the world we've already processed our thoughts and accept that we're trans, family and friends are just discovering it. We come out and often we're ready to be making changes in our lives. It's all "hey, I'm trans, I want to be referred to as this, dress like that, go on hormones and have surgery (delete as applicable)". There's no build up to it. It's right there in their face, and they're pressured into getting on with it and coping straight away.
And they don't know about the struggles we've been through, they don't feel the same pain as us, and they don't understand dysphoria. How many of us have self-harmed or attempted/idealised suicide without ever saying a word? I know for me it's not something I'm proud of, or talk about, to anyone. Some people know bits and pieces, but that's because I cope by making a joke out of everything. I was seeing a mental health team, did anyone outside of those meetings know why, that I was in a deep bout of depression, self-harmed and attempted suicide more times than I can remember? They knew anecdotes of it, they saw the evidence, but never knew the pain because I laughed and said that I was off to get my strait-jacket fitted. I wasn't telling them the truth. They don't know our pain and they don't understand why we can't just be masculine/feminine people. Dysphoria isn't something they know the pain of, and how it's do or die.
It's no harder for them than it us, but it's harder in a different way, and I think that's the way people are referring to when they say that. It's harder for them to accept than us, because it's a shock, but it's not harder in terms of the mental toll it takes.