What is the need of Tawasol?
It is known that there are places that Allah loves to be worshiped in:
[Such niches are] in mosques which Allah has ordered to be raised and that His name be mentioned therein; exalting Him within them in the morning and the evenings [24:36]
In addition, we are sinful; Our sins are taking us away from Allah so we need a "mean" to come back to Allah and ask his forgiveness. This mean must be a sinless person i.e. the prophet and his family.
Is Tawasol allowed?
to know if Tawasol is allowed, we should check if the Islamic sources and Qur’an is -obviously- our first reference:
O you who have believed, fear Allah and seek the means [of nearness] to Him and strive in His cause that you may succeed.5:35
In this verse, Allah is asking us to seek the means to him. It uses the Arabic word "الوسيلة" which proves that Tawasol is allowed and even encouraged.
They said, "O our father, ask for us forgiveness of our sins; indeed, we have been sinners."
In this verse, the sons of Jacob asked him to ask Allah to forgive their sins. Note that they could have asked Allah directly.
There are plenty of other verses and Ahadith that describes that Tawasol is allowed.
Tawasol between Shia and Sunni
It is wrong to think that only Shia allow Tawassol. Sunni allow Tawassol too. The difference is that Sunni don't allow Tawassol with the dead while Shia do.
Why do Shia allow Tawassol with the dead?
The following verse:
( وَلَو أَنَّهُم إِذ ظَلَمُوا أَنفُسَهُم جَاءُوكَ فَاستَغفَرُوا اللَّهَ وَاستَغفَرَ لَهُمُ الرَّسُولُ لَوَجَدُوا اللَّهَ تَوَّاباً رَحِيماً
And We did not send any messenger except to be obeyed by permission of Allah. And if, when they wronged themselves, they had come to you, [O Muhammad], and asked forgiveness of Allah and the Messenger had asked forgiveness for them, they would have found Allah Accepting of repentance and Merciful.
The debate here is about this verse. Sunni's say that this verse is just for the past "اذ" proves that the whole verse should be in the past tense.
Shia say that the verb "جاءوك" came within a conditional which adds generality to the verse. Here is a Shia source and a Sunni one. Since the debate is about the Arabic language itself, it is hard to explain it here in English