I just started reading The Odyssey, and Telemachus is mad because suitors of his mother Penelope are staying at his place and eating all his food. Do they have a right to be there?
Well this isn't an historical question so much as a literary one. I'm tempted to say you should just keep reading and it will become clear, but I've been through Homer enough times I know people don't always catch all the plot points without having them flagged.
There's two hundred years of scholarship on trying to understand what Homer's relationship to history was, and more on top of that about how we should understand the literary aspects of his text.
I'm I think what M. West calls a neo-analyst; I think Homer's poems are more the product of a tradition than a single artistic mind (though there is a single artistic mind who had the tradition for raw material); I'm very skeptical of Homer's information about the Mycenaean World; I even lean to a Peisistratid date for getting written down. Basically I don't think there's much historical here to discuss.
So you're right about Telemachus' complaint about the suitors destroying the wealth of his house. This is the complaint he voices to Athena in disguise as Mentes in the first book. But the suitors have a defense. The first plan Athena proposes is relatively conflict free. Telemachus should go to the assembly and tell the suitors to go home, and his mother Penelope, to get married again, should go to her father's home, and the suitors should pick up their case there.
Alright, so that's the point of contention. Penelope hasn't gone back to her father's place like the characters expected, and so they occupy Telemachus' place. What's supposed to happen, as far as the characters are concerned, has been completely ignored.
Telemachus in book two makes the case that Athena suggested in the assembly; the suitors should go to Penelope's father's house and press their case there.
Antinoos speaks up though; defending the suitors' position. Penelope is leading the suitors on, not getting married on purpose, and he demands that Penelope should be sent to her father's house.
So Athena basically said what's up. Penelope should go to her father's house, and the suitors should court there. Telemachus just asked the suitors to take up the issue with Penelope's father. The suitor who speaks responds that Penelope should also go to her father - which is what Athena suggested.
So, no one's right. The whole situation is unusual and all three parties, Telemachus, the suitors, and Penelope, are doing their best to work it out toward their own ends.
I had to read Homer three or four times before I started putting the plot together, and I've read a lot of it in Greek too. Keep at it. Homer's wonderful once you get enough to understand what's going on, but it's a steep curve.