What Are the Days of Tashriq?
Question

What are the days of Tashriq? What is special about them and what distinguishes them from other days?
Praise be to Allah.Virtues of the Days of Tashriq
The days of Tashriq are the 11th, 12th and 13th of Dhul-Hijjah . There are several verses and ahadith which speak of their virtue: 

1 – Allah says (interpretation of the meaning): 
“And remember Allah during the appointed Days.” [al-Baqarah 2:203]
These are the days of Tashriq. This was the view of Ibn ‘Umar and most of the scholars. 
2 – The Prophet (peace and blessings of Allah be upon him) said concerning the days of Tashriq: “They are days of eating, drinking and remembering Allah.” 
The Prophet (peace and blessings of Allah be upon him) forbade fasting on these days : “Do not fast on these days, for they are the days of eating, drinking and remembering Allah.” Narrated by Ahmad, 10286; classed as sahih by al-Albani in al-Silsilah al-Sahihah, 3573.
Dhikr during the days of Tashriq
Dhikr (remembering Allah) is enjoined during the days of tashriq in several ways: 

Remembering Allah immediately after the prescribed prayers by reciting Takbir. This is prescribed until the end of the days of Tashriq according to the majority of scholars.
Remembering Him by saying Bismillah and Allahu akbar when slaughtering the sacrificial animal. The time for slaughtering the hadiy and udhiyah lasts until the end of the days of Tashriq.
Remembering Allah when eating and drinking. It is prescribed when eating and drinking to say Bismillah at the beginning, and to praise Him (say Al-hamdu Lillah) at the end. According to the hadith narrated from the Prophet (peace and blessings of Allah be upon him): “Allah likes His slave when he eats something to praise Him for it, and when he drinks something to praise Him for it.” (Narrated by Muslim, 2734)
Remembering Him by saying Takbir when stoning the Jamarat on the days of Tashriq. This applies only to the pilgrims on Hajj.
Remembering Allah in general. It is mustahabb to make a lot of dhikr during the days of Tashriq. ‘Umar used to recite Takbir in Mina in his tent, and when the people heard him they recited Takbir too and Mina echoed with the sound of their Takbir. Allah says (interpretation of the meaning): 

“So when you have accomplished your Manasik [rituals of Hajj], remember Allah as you remember your forefathers or with a far more remembrance. But of mankind there are some who say: “Our Lord! Give us (Your Bounties) in this world!” and for such there will be no portion in the Hereafter.
And of them there are some who say: “Our Lord! Give us in this world that which is good and in the Hereafter that which is good, and save us from the torment of the Fire!”” [al-Baqarah 2:200-201]
Many of the salaf regarded it as mustahabb to make a lot of du’a during the days of tashriq. 
The words of the Prophet (peace and blessings of Allah be upon him), “These are days of eating, drinking and remembering Allah” indicate that eating and drinking during the days of ‘Eid are means that help one to remember Allah and obey Him; perfect gratitude for the blessing means using it to help one obey and worship Allah, 
In His Book Allah commands us to eat of good things and thank Him for them. So whoever uses Allah’s blessings to commit sin is showing ingratitude for the blessing of Allah, so he deserves to have it taken away from him. 
Reference: 
Lataif al-Ma’arif by Ibn Rajab, p. 500.
And Allah knows best.