Within Islām, sending Ṣalawāt (salutations) upon The Messenger of Allāh ﷺ is one of the commands enjoined upon us by Allāh, as stated in The Qurʾān:
إِنَّ ٱللَّهَ وَمَلَٰٓئِكَتَهُۥ يُصَلُّونَ عَلَى ٱلنَّبِيِّۚ يَٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُواْ صَلُّواْ عَلَيۡهِ وَسَلِّمُواْ تَسۡلِيمًا
Indeed, Allāh confers blessing upon The Prophet, and His angels [ask Him to do so]. O you who have believed, ask [Allāh to confer] blessing upon him and ask [Allāh to grant him] peace.
– Sūrah al-Aḥzāb, ʾĀyah 56
Doing so contains tremendous virtues as mentioned by The Prophet ﷺ himself within Ṣaḥīḥ Aḥādīth such as:
Whoever sends salutations once for me, Allāh blesses him ten times & removes ten errors from him.Ṣaḥīḥ Muslim 1/288, al-Adab al-Mufrad #643
Among the best of your days is Jumuʿah. On that day, pray to Allāh to exalt my mention frequently, for your such supplications are presented to me.Riyāḍ aṣ-Ṣāliḥīn #1158, Sunan Abū Dāwūd #1531
Shaykh Ṣāliḥ al-Fawzān حفظه الله explains: “It has been reported that the meaning of Allāh’s Ṣalāh, Most High is He, is His praise of him before the Angels, & the Angels’ Ṣalāh means supplication (for him), & peoples’ Ṣalāh means al-Istighfār (seeking forgiveness for him).
Allāh, The Glorified had informed in this ʾĀyah of the station of His Prophet amongst the highest companion, that He praises him in the presence of the closest Angels, & that the Angels supplicate for him. Then He, The Most High, ordered all the inhabitants of the lower world (earth) to make Ṣalāh (seek graces) upon him & salute him with peace in order that praise upon him by the inhabitants of the higher world (heavens) & the lower world (earth) coincide.
The meaning of making peace upon him is that you should salute him with the greeting of Islām. So, whenever one seeks Ṣalāh upon The Prophet ﷺ, one should combine Ṣalāh & Taslīm (saluting him). He should not restrict it to one of the two. He should not say: ‘Ṣallallāhu ‘alayhi’ only, nor should he say ‘Alahyi as-Salām’ only, because Allāh had ordered both.” 
Shaykh Muḥammad ibn Ṣāliḥ al-‘Uthaymīn رَحِمَهُ ٱللَّٰهُ (d. 1421 AH/2001 CE) also explains: “The Ṣalāh from Allāh upon His Messenger does not mean what some of the people of knowledge conceive, that the Ṣalāh from Allāh (to him) is mercy & that from the Angels is asking for forgiveness (for him) & the one from humans means supplication. This is not correct at all. Allāh’s Ṣalāh upon an individual is His praising him in the highest gathering as explained by Abū al-ʿAlīyah & corroborated by the leading researching scholars.
Therefore, whoever sends Ṣalāh upon Muḥammad once; Allāh will exalt his mention ten times in the highest gathering. This is undoubtedly a tremendous favor.” 
May Allāh protect Shaykh Ṣāliḥ al-Fawzān & may Allāh have mercy upon Shaykh Muḥammad ibn Ṣāliḥ al-‘Uthaymīn.
All praise is due to Allāh, may His peace & blessings be upon our final Prophet Muḥammad, his family, his companions, & all those who follow him in guidance.
 ʿAqīdah at-Tawḥīd (عقيدة التوحيد) by Shaykh Ṣāliḥ al-Fawzān | Page 225
 al-Qawl al-Mufīd Sharḥ Kitāb at-Tawḥīd (القول المفيد شرح كتاب التوحيد) by Shaykh Muḥammad ibn Ṣāliḥ al-‘Uthaymīn | Volume 1, Page 458
Ṣafīullāh Labīb ibn Salīm ʿAbd al-Malik
23rd of Dhū al-Hijjah 1443