It is said that if one slaughters (sacrifices) an animal for himself/herself or anyone else, then on the Day of Resurrection this animal will help him pass over (sitting on its back) the bridge of Hell. I can't understand the reason for such claims and could you give a fatwa on this?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

There are some narrations about what you mentioned in your question.

Ad-Daylamy (a scholar of Hadeeth) mentioned a narration from Abu Hurayrah with a weak chain of narrators that reads: “You should fatten your sacrificing animals as (in the Hereafter) they will be your riding animals on the path (the bridge)”.

Imam Ibn as-Salah said about this Hadeeth: “This Hadeeth is not known and it is not authentic to our knowledge.”

Imam Ibn al-‘Arabi mentioned in his explanation of Sunan at-Tirmithi that “There is no authentic Hadeeth describing the excellence of sacrifice; even the Hadeeth stating they will be your riding animals to Paradise is not authentic.”

This position is also held by the two eminent Imams Ibn Hajar in his book at-Talkhees al-Habeer and al-'Ajlooni in his book Kashf al-Khafaa.

You know now from the above that there is no sound Hadeeth from the Prophet regarding what you mentioned in your question.

So, just sacrifice as it is recommended by the Sunnah and you will get its great reward, Allah willing.

Allah knows best.