This is way too complex an issue to be covering in a forum post and I'm possibly not the best person to do it but I'll try and point you in the right direction.
When people talk about Labour anti-semitism, or indeed anti-semitism in general, they are bundling in anti-zionism. Not only that they are extending anti-zionism to include any criticism of the state of Israel.
What's the difference between anti-Semitism and anti-Zionism?
For me that's a big part of the problem. To the man on the street with little interest in politics if you scream the headline Labour anti-semitism they imagine a Labour MP being anti-jew or even worse a Holocaust denier. They don't think its a Labour member who pays a few quid a month tweeting that Israelis shouldn't bomb Palestinians.
So then you're into the Israel - Palestine issue which has been going on for decades. This is a few years old now (2016) but a decent summary.
Now add in the Board of Deputies. They present themselves, and the majority of the media facilitate this, as the voice of all Jews. That's clearly nonsense and there's other Jewish groups with very different views but that doesn't matter when they have the ear of the media. They are pro-Zionism and are a key part of the Zionist Lobby in the UK (
Israel lobby in the United Kingdom - Wikipedia).
They took issue with Corbyn as Labour leader as he has long been a supporter of Palenstine's right to exist and the 1967 borders, basically the borders before the six day war. That's not a controversial view, Obama gave speeches saying the same thing and the UN security Council has long stated that Israel’s settlements in Palestinian territory since 1967 are illegal and break international law. But that was when the claims of Labour anti-semitism went into overdrive and the current board leader, van Der Zyl, has pushed the idea that anti-zionism, or any criticism of Israel, is anti-Semitic.
If that wasn't enough enter Donald Trump! He's basically gone against the rest of the international community and stirred things up to the point that an Israeli invasion is expected any day now. Be interesting to see how Starmer, who pushes his human rights background, approaches that without upsetting the Board of Deputies.
Why Trump and Jerusalem was not about peace
Trump's Mideast plan gives Israel a green light. Here's what that looks like - CNN
Bet you wish you hadn't asked! Might well be others on here who can explain it better than me.