Short answer
Islam stresses remembering God (﷾) because sin usually comes when God (﷾) is forgotten. While a person keeps God (﷾) in mind, wrongdoing is hard. Satan works to fill the mind with other things so that God (﷾) is forgotten. So the acts of worship are built to keep God (﷾) in mind. The five daily prayers are spread through the day, between meals, work and sleep, and the Qur'an says prayer is for remembering God (﷾). Fasting trains self-restraint. Alcohol and gambling are banned partly because they block remembrance. When a believer does sin, remembering God (﷾) brings him back to ask forgiveness. Islam asks for moderation in most things, but it asks for the remembrance of God (﷾) in abundance. The Qur'an adds a promise: in the remembrance of God (﷾), hearts find rest.
Full explanation
Forgetting God (﷾) is how sin begins
The Qur'an says Satan wins people over by making them forget the remembrance of God (﷾). It says those he has won are his party, and that they are the losers. It names intoxicants (anything that clouds the mind, such as alcohol) and gambling as his tools. They stir up hatred and turn people away from remembering God (﷾) and from prayer. This is one of the reasons the Qur'an gives for forbidding them.
“The Evil One has got the better of them: so he has made them lose the remembrance of Allah. They are the Party of the Evil One. Truly, it is the Party of the Evil One that will perish!”
“Satan's plan is (but) to excite enmity and hatred between you, with intoxicants and gambling, and hinder you from the remembrance of Allah, and from prayer: will ye not then abstain?”
Worship is built for remembrance
In the Qur'an, God (﷾) tells Moses to establish prayer for His remembrance. The Arabic word is dhikr, which means remembrance. Yusuf Ali, whose English translation of the Qur'an this site uses, renders it "praise" here. Five times a day, prayer breaks into eating, work and sleep to feed the soul. It also holds a person back from shameful deeds. Fasting is prescribed so that believers learn taqwa (self-restraint and mindfulness of God (﷾)).
“Verily, I am Allah: There is no god but I: So serve thou Me (only), and establish regular prayer for celebrating My praise.”
“Recite what is sent of the Book by inspiration to thee, and establish regular Prayer: for Prayer restrains from shameful and unjust deeds; and remembrance of Allah is the greatest (thing in life) without doubt. And Allah knows the (deeds) that ye do.”
“O ye who believe! Fasting is prescribed to you as it was prescribed to those before you, that ye may (learn) self-restraint,-”
Remember God (﷾) often
Islam asks for balance in food, sleep, wealth and even worship. Remembrance is the exception. The Qur'an tells believers to remember God (﷾) much. The Arabic is udhkuru Allaha dhikran kathira, which Yusuf Ali renders "celebrate the praises of Allah, and do this often". It also promises that God (﷾) remembers those who remember Him. The Prophet (ﷺ) likened the one who remembers his Lord to the living, and the one who does not to the dead.
“O ye who believe! Celebrate the praises of Allah, and do this often;”
“Then do ye remember Me; I will remember you. Be grateful to Me, and reject not Faith.”
Abu Musa reports that the Prophet (ﷺ) said the one who remembers his Lord, compared with the one who does not, is like the living compared with the dead.
The Prophet (ﷺ) reports that Allah says: I am as My servant thinks of Me, and I am with him when he remembers Me; if he comes a hand-span nearer to Me, I come an arm's length nearer to him, and if he comes to Me walking, I go to him running. Also in Sahih Muslim 2675.
Remembrance brings the sinner back
Everyone has moments of weakness. A person with no means of remembering God (﷾) can sink deeper with every sin. The Qur'an praises those who, after doing wrong, bring God (﷾) to mind and ask His forgiveness. It also promises that hearts find rest in His remembrance.
“And those who, having done something to be ashamed of, or wronged their own souls, earnestly bring Allah to mind, and ask for forgiveness for their sins,- and who can forgive sins except Allah?- and are never obstinate in persisting knowingly in (the wrong) they have done.”
“"Those who believe, and whose hearts find satisfaction in the remembrance of Allah: for without doubt in the remembrance of Allah do hearts find satisfaction.”
Perspectives
The Islamic perspective
Remembering God (﷾) is the purpose running through all worship. It guards against sin, which grows when God (﷾) is forgotten, and it turns the sinner back to repentance. The Prophet (ﷺ) likened a life without it to death.
The Qur'an says prayer is established for the remembrance of God (﷾) and that it restrains from shameful deeds.
Fasting is prescribed so that believers learn self-restraint.
The Qur'an commands believers to remember God (﷾) often, and promises that God (﷾) remembers those who remember Him.
The Qur'an says Satan makes people forget the remembrance of God (﷾). Intoxicants and gambling are his means to that end.
Sin is most often committed when God (﷾) is forgotten. So keeping God (﷾) in mind is the first guard against it.
The Qur'an praises those who remember God (﷾) after doing wrong and ask His forgiveness.
The Prophet (ﷺ) likened the one who remembers his Lord to the living. He likened the one who does not to the dead.
The Qur'an promises that hearts find rest in the remembrance of God (﷾).