These things many not be fully related to official scriptures but mainly folklore. Temple history etc.
The Snana Yatra (Odia: ସ୍ନାନ ଯାତ୍ରା) is a bathing festival celebrated
on the Purnima (full moon day) of the Hindu month of Jyeshtha. It is
an important festival of the Jagannath devotees. It is the birth day
of lord Jagannath.This is the first occasion in the year as per the
Hindu calendar, when the deities Jagannath, Balabhadra, Subhadra,
Sudarshan, and Madanmohan are brought out from the Jagannath Temple
(Puri) and taken in a procession to the Snana Bedi. There they are
ceremonially bathed and decorated for a public audience with the
devotees.
It is a belief among devotees of Lord Jagannath that if they make a
pilgrimage to see the deity on this day, they would be cleansed of all
of their sins. Hundreds of thousands of devotees visit the temple on
the occasion. The Skanda Purana mentions that King Indradyumna
arranged this ceremony for the first time when the idols of the
deities were first installed.
On the eve of the Snana Yatra (which means the Bathing festival, in
Sanskrit), the idols of the deities are brought out in a grand
procession from the garbhagriha (sanctum sanctorum) to the Snana Bedi
(bathing platform). Devotees come to view the deities.
On the day of the Snana Yatra, the deities are bathed with 108 pots,
of ritually purified water drawn from the northern well of the temple
to the accompaniment of religious incantations. In the evening, at the
conclusion of the bathing ritual, Jagannath and Balabhadra are dressed
up in elephant headgear representing the God Ganesh. This form of the
God is called the 'Gajavesha'.
After the Snana Yatra the Gods are traditionally believed to fall ill
and are kept in a sick room to recuperate in privacy under the care of
the Raj Vaidya. During this period known as Anasara the Gods cannot be
seen by devotees. At this time three pata chitra paintings are
displayed for devotees to view instead It is said that with the
Ayurvedic medication ('pnachan') administered by the Raj Vaidya the
Gods recover in a fortnight and resume giving an audience to their
devotees. (Wiki)

Until amavasya Lord Jagan natha, brother Balabhadra, and sister Subhadra will undergo fever. Next Shukla paksha pratipada they will get well. After this, they will ride their ratha on Shukla Dwitiya.
The period of illness is known as "anasara". They will be given special herbs like Dasa-moola and special diet during the illness.
They will be offered a special decoction of black pepper, cinnamon, fennel, liquorice etc. Source
Bells, mridangam etc. will be stopped in order to give rest to the lord.
Different oils and resins along with musk will be applied to the vigrahas for rejuvenation.
No darshan will be allowed in the temple till the shukla pratipada of Ashadha. Some priests will do pooja secretly.