It is very simple. First of all, Islam is the only religion with the Book that clearly guides, the Qur'an. It is not changed since its beginning, it is clearly from Allah as so many verses of it point out and it contains lessons in so many areas of spirituality that it leaves nothing out to guide Muslims with. The Qur'an is the unadulterated speech of Allah. It is protected by Allah Himself (15:9).
Secondly, the Holy Prophet of Islam (saw) was the Prophet of Allah, for whom history records his presence with more accuracy than is known for any other Prophet of God. Even those who claimed to be Prophets from many other religions after him, there is no account like his and no large set of sayings recorded with an intricate science for transmission like it has been done for him. His words have the best guidance, and Muslims for the past 1400 years still derive very important principles of guidance from them. It speak to all of us Muslims now as it did 1400 years ago.
Thirdly, most importantly, the concept of Allah is so clear and purely Divine that any and every religion falls short of coming up with the proper understanding of Allah. He is ineffable, and yet we know of His attributes. We know Him as clearly as the human heart can understand. What we know of Him speaks to everyone whether they be religious or irreligious. One only need to look for Allah's attributes and then see how clearly Islam has come to know of how Allah is. We can't know very much, but what we know of Him is acceptable to anyone and everyone. Only those who fail to read and understand on His attributes foolishly speak against them.
A fourth reason why Islam is correct and other religions are wrong, is the fact that the signs mentioned in the Holy Qur'an and the sayings of the Prophet of Allah, Muhammad (saw), are consistently true with every age that passes. We see signs all the time, and there is no shortage. We do not need any more miracles like the sea being split into two, water gushing from a stone, a burning bush, a blind person's eyesight being restored, etc. Our signs are all contained in the Holy Qur'an, which is the perfect book, and those who learn and know Arabic always see amazing signs in the Qur'an regarding its linguistic miracles. When a Muslim sees these, their faith is always anew and fresh and they are ready to be even more fervent in their faith to Allah.
These are four reasons which can be detailed with over a thousand facts for each that prove their authenticity. As for signs, there are thousands of them in the Qur'an alone, not to mention the amazing signs in the sayings of the Holy Prophet Muhammad (saw). These are always beneficial for establishing the unarguable fact that Allah exists and is always with us. Besides that, the Holy Prophet Muhammad (saw) did do hundreds of miracles. What he did amounts to more than what is recorded of every Prophet (as) from the past combined.
As for why other religions are wrong, I cannot answer that completely. That is a very loaded question because it would require me to answer on behalf of all of those religions. I can state the qualities Islam has which other religions do not have. I mentioned above some of these, but the other very important fact is that Islam has a perfect code for life. Jewish law is in itself very much detailed and good in its own way, but it is not the perfect law. Islam is the religion with the perfected law of Allah, which is for mankind until the time that the Day of Judgment is established. Even when verse 5:3 was revealed, which stated that Islam is now the perfect religion, the Jews said they would have celebrated a festival if they had the same revelation sent by Allah to them. This constitutes a proof that despite any other religion's set of laws, the Holy Qur'an is the only Book which is indisputably from Allah Himself, that says such a statement as in verse 5:3, that the law of Allah is completed unto mankind. After Islam, no other Sharia can come from Allah, and all other Sharias are contained in Islam and it is the best Sharia for mankind. No man-made law can ever supersede this, and even men try to do so, they will not succeed because Allah does not allow anyone to share in His kingdom, especially those who go astray.
... I have a feeling you will grapple with it.