Help me install my dishwasher

Does the leak originate at the connection between supply and line hose, or between the line hose and the dishwasher? My guess is that you haven't tightened the connections, you have used plumber's tape on a joint that doesn't need it, or you have a faulty line. But there are too many unknowns to give a definitive answer. However, you definitely do not want to remove or puncture anything from the dishwasher innards, which will result in an expensive repair. Call in outside help who can look at the situation in-person.

It's most likely a screen/washer to catch sediment.

If you can post close-up photos of the hose end and the dishwasher connection point, it will probably be obvious to somebody what's wrong. Most likely cause of gross leaks in a screw-on hose fitting is a lost rubber sealing washer.

A good size beach towel is going to take care of most of your leak problem while you get this fixed and are turning the water off and on (mostly off) to see if you have solved the problem. (IE: make changes; have someone observe the underside of the dishwasher, turn on the water and have them check for leaks. Even a hose that blows completely off is only going to spill a litre of two of water while you turn the supply tap off that you are standing right next to.) Wring it out as needed. Like flabdablet said if your hose is the right one for the dishwasher then the problem is 99.5% either a defective hose or sealing washer. Because there was no power to the D/Washer the leak should be confined to down stream of the fill valve (IE: the part the hose screws onto) (these are normally closed so that every power outage doesn't cause a kitchen flood). Sometimes the screen on the fill valve inlet will get crudded up (even right from the manufacturer) which will cause your D/W to not fill but won't cause a leak. So check the hose has a quarter sized flat ring of rubber in the male part of the threads. And ya, even a crappy cell phone pick of the general assembly and a view down the interior of of the valve and hose would really help pin this down. If your washer does not have a connection with course threads exactly like a exterior water tap then you don't have the right hose and you should either call someone or post a few pics.

"I immediately disconnected the water supply, and examined the water inlet in the dishwasher. I put my finger in it to see if there was something inside it, and there seems to be something (some kind of membrane or plastic piece) in the inlet. " Hmm... on third reading it appears you are talking about the inlet actually within the "tank" of the Dishwasher (IE: where the dishes go). If that is the case you are looking in the wrong place for a leak as 99.999% of the time water can't reach that point until after you energize power to the dishwasher and actually turn it on (and even then not all the time). And that bit is supposed to be springy.

Have you read the installation instructions? Maybe start there. If you don't have a copy, you should be able to obtain something at the manufacturer's website.

Are you sure that's the proper inlet? All the dishwashers I've installed [OK, three, but still] have had a https://www.google.com/search?rlz=1C1CHFX_enUS457US457&sourceid=chrome&ie=UTF-8&q=dishwasher+water+supply+line#q=dishwasher+water+supply+line&hl=en&rlz=1C1CHFX_enUS457US457&prmd=imvns&source=univ&tbm=shop&tbo=u&sa=X&ei=n5T8Tvi1Nsi9gAetle1X&ved=0CHsQrQQ&bav=on.2,or.r_gc.r_pw.r_cp.,cf.osb&fp=2279860aca39e76e&biw=1166&bih=636. Are you sure you aren't connecting the water supply to the drain line? Is this a new dishwasher? As mentioned, do you have the install instructions? If not, given the model number, you can probably find the install instructions online. I wouldn't go poking things further into supply lines without a very clear idea of what's what. It's extremely unlikely that connecting a water supply line to a dishwasher involves pushing a bit of plastic back into the inlet valve assembly.

The larger course thread connector is typical on some European brands and european import badge engineered models like some whirlpools.