Ibn Kathir introduced this saying:
Al-Bukhari said: "Abu Al-'Aliyah said: "Allah's Salah is His praising him before the angels, and the Salah of the angels is their supplication." Ibn 'Abbas said: "They send blessings." Abu 'Isa At-Tirmidhi said: "This was narrated from Sufyan Ath-Thawri and other scholars, who said: 'The Salah of the Lord is mercy, and the Salah of the angels is their seeking forgiveness. (Source qtafsir)
Ibn Hajar al-'Asqalani considerd the interpretation given by Abu al-'Aliyah أبو العالية الرياحي as the best on this topic.
This gives an answer about the kind of blessings and prayers. Unfortunatley the authors of qtafsir didn't go further in their translation and didn't address the deeper meaning:
(**Note whenever I quote Arabic text the English translation is of my own and should be taken with care!)
والمقصود من هذه الآية : أن الله سبحانه أخبر عباده بمنزلة عبده ونبيه عنده في الملأ الأعلى ، بأنه يثني عليه عند الملائكة المقربين ، وأن الملائكة تصلي عليه . ثم أمر تعالى أهل العالم السفلي بالصلاة والتسليم عليه ، ليجتمع الثناء عليه من أهل العالمين العلوي والسفلي جميعا . (Tafsir ibn Kathir)
And what is meant by this verse is the following: Allah Exalted is He informed in this verse his servants about the high rank of his servant and prophet with Him and His exalted assembly (of angels). That he praises him with his favourite (closest) angels and that the angels send prayers to him. Then He Exalted is He ordered the people of the lower world to send prayers and blessings to him. To gather both praising from the lower and upper world together.
That's why Allah() says:
And raised high for you your repute (94:4)
So the prayers and blessings mean:
“Blessings upon the Prophet (peace and blessings of Allaah be upon him)” – according to the majority of scholars, what is meant is mercy from Allaah, prayers for forgiveness offered by the angels, and du’aa’ offered by humans. (Source: islamqa #69944)
Imam al-Qurtobi -in his tafsir- introduced the verse similarly to ibn Kathir saying:
هذه الآية شرف الله بها رسوله عليه السلام حياته وموته ، وذكر منزلته منه ، وطهر بها سوء فعل من استصحب في جهته فكرة سوء ، أو في أمر زوجاته ونحو ذلك . والصلاة من الله رحمته ورضوانه ، ومن الملائكة الدعاء والاستغفار ، ومن الأمة الدعاء والتعظيم لأمره .
This verse is a honouring of Allah for his Prophet peace be upon him in his life and death and a mentioning of his rank with Allah, and is a cleaning for every evil deed of those who carry any idea of evil towards him, or about his wives (*) and likewise, and the prayer from Allah is (His) mercy and (His acceptance and from the angels is supplication (du'a) and seeking forgivness and from the Ummah is supplication and high esteem for him.
(*) Allah mentioned the wives of the prophet in the context of this verse, in the verses before and later! Note that the same applies to verse (33:33) which calls them ahl al-bayt.
Finally note that Arabic world for prayer الصلاة (as-Salah) has a link to the word الصلة (as-Siallah) the connection. And via du'a we are in connection with Allah: We make a supplication and he listen and answers it.
Maybe adding the word "benefit" was a bit misleading, of course Allah doesn't benefit from praising any of his creations be it our prophet () nor actually do the angels as they are following the orders of Allah.
As to the why I think it is clear that Allah's praising means that our prophet has a high rank with Allah.
As to the how it is explained in the couple of quotes above.