I think most baby monitors work with radio signals. Those would be very easy to hack: get close enough with a radio device that is tunable to the right frequencies, then scan through frequencies until you hear the signal. I'm sure you could look up somewhere what frequencies are allocated to such devices. Then someone with a little skill in electronics could modify an ordinary radio to receive the proper frequencies.
As far as I know there's no way to tell who has listened to a radio signal. There's no electronic trace that would be left.
The main trick I see would be that you'd have to get close enough to pick up the signal. These devices deliberately have a short range -- so that they don't interfere with each other. I just checked some ads and saw advertised ranges generally around several hundred feet. You might get caught because someone saw your car sitting in front of their house or that sort of thing. If you're talking about a prince, he likely has a big piece of property, so getting close enough to the house to pick up a signal might be an issue. If you have to break into the grounds to get close enough to the transmitter, obviously that creates all sorts of dangers of getting caught.
Some baby monitors encrypt the signal, presumably exactly to prevent others from listening in like this. In that case, you've have to somehow discover the encryption key. I don't have any idea how sophisticated the encryption is, so I can't say how hard it is to crack. Someone like a prince, who presumably has valid fears of kidnapping or assassination or just paparazzi, might well get a baby monitor that encrypts the signal. I'd assume that to a prince the extra cost would be a trivial issue.
If the baby monitor is an internet device, it's a whole different story. I don't know what sort of security they have on internet baby monitors, whether they have passwords or whatever. But let's just talk about hacking internet devices in general. There are basically three ways (I can think of) to hack it.
Method one: Connect to his network. If it's a Wifi network, you can make the physical connection from outside the house as long as you're in range of the signal. If the prince has no security on his wifi, this would be trivial. Hopefully he's not that foolish. Assuming he has a password or some such to connect to his network, you'd have to crack that.
If you gain access that way, his router should keep some record of accesses. Some routers keep long logs, but any router will keep track of any device that connected to the network. So if they checked, they'd know that someone connected. They'd know what IP his router assigned to you, but that's useless information. They'd know the machine ID of the device you used to connect. If someone had a registry somewhere of what machine IDs were sold to who, maybe they could track you down. I don't know if anyone keeps this information.
If he doesn't have a wifi network but relies on wired connections, you'd have to physically connect to his lines. That presumably involves breaking into his house.
Method two: Connect over the Internet. This assumes that the baby monitor is accessible from outside the local network, but I guess if you're going to have an internet baby monitor, the idea is so that you can check on the baby while away from home.
In that case you again have to gain access to his network, presumably enter some passwords or something. In this case his ISP and your ISP will have logs of the connection. They can track down your IP and probably determine your physical address. I presume that clever criminals have come up with ways to beat this: run messages through long, hard-to-trace routes or some such. There are "anonymizer services" created to let people hide their internet identity. The service would know who you really are, and if they're subpoenaed I presume they'd have to turn over the information. But if they're not in the same country, maybe they'd just say no.
Method three: Sniff the lines. That is, intercept the signals travelling over the internet and listen in. I don't know how to actually do this, but I know it's done. Presumably if you do it right you leave no trace, because, like listening in on a radio signal, you're not sending any messages, you're just listening to someone else's messages.