When it comes to altering one's body, sometimes it's allowed:
It should be understood that with regard to changing the creation of Allah, some of it is permissible and allowed, such as circumcision, shaving the pubic hair, plucking the armpit hair, and removing defects and deformities.
and sometimes it's not allowed. In the case of tattoos, there's ahadith which imply they are forbidden:
Narrated Ibn `Umar: Allah's Messenger said, "Allah has cursed such a lady as lengthens (her or someone else's) hair artificially or gets it lengthened, and also a lady who tattoos (herself or someone else) or gets herself tattooed.
Sahih al-Bukhari 5937 (see also Sahih al-Bukhari 5946)
As to why tattoos are forbidden, altering one's body is not a definitive reason.
Islam Q&A include "changing the creation of Allaah" as a reason:
Tattooing... is haraam in all forms, whether it causes pain or not, because it involves changing the creation of Allaah, and because the Prophet (peace and blessings of Allaah be upon him) cursed the one who does tattoos and the one for whom that is done.
As does IslamWeb:
Among the wisdoms that the scholars may Allah have mercy upon them have mentioned is that having tattoos is considered as changing the creation of Allah, and in some narrations the following addition has been mentioned "…whereby they change Allah's creation."
Such ahadith include Sunan an-Nasa'i 5108 and one in Jami` at-Tirmidhi.
But there are other reasons offered: unnecessary pain (IslamWeb), and it constitutes filth (Qibla.com, sourced from IslamQA.org). And when asked about the reasons behind tattooing being haram:
Q: ... The two most common reasons I heard is ghusal will be invalid because it covers the skin and because it is permanently changing the body. ...
A: ... the impermissibility is not based on what you have mentioned.