Why does Chinna Jeeyar Swami carry a bunch of sticks and a white cloth or flag attached to them? What is the significance of these sticks? Why isn't it just a single stick; why should it be a cluster of sticks? Why are these sticks sometimes covered in cloth?
Not just Chinna Jeeyar Swami, but all Sri Vaishnava Acharyas carry these three sticks. They are called the Tridanda, and they are carried by those who have taken up Tridandi Sanyasa, which is the type of Sanyasa that Sri Vaishnavas follow. The appearance of Sanyasis in general is described in this chapter of the Manu Smriti:
His hair, nails, and beard being clipped, carrying an alms-bowl, a staff, and a water-pot, let him continually wander about, controlling himself and not hurting any creature.
But the number of sticks that the staff is made of varies based on the type of Sanyasa. Advaitins follow Ekadanda Sanyasa, where they carry only one stick to symbolize their belief that only Brahman exists. Sri Vaishnavas, on the other hand, follow Tridandi Sanyasa, where the three sticks symbolize the three entities which exist according to Vishishtadvaita: Chit or souls, Achit or inanimate matter, and Ishwara, the supreme lord. Here's what this Sri Vaishnava website says:
In our Srivaishanava Sampradayam, Pontiffs carry triple staff (tridandam) made out of three small wooden sticks. This is to signify the three components of our philosophy - chit, achit and easwara. It is mandatory for our Pontiffs to carry tridanda always with them. It is also said that "Vishnuroopam Tridandagyam" i.e., Tridandam is Vishnu's roopam.
But Sri Vaishnavas aren't the only Tridandi Sanyasis; this book says that the Vedantic philosopher Bhaskara, who believed in the philosophy of Aupadhika Bhedabheda as I discuss here, was a Tridandi Sanyasi. And Gaudiya Vaishnavas also believe in Tridanda Sanyasa, as described in this letter by Srila Prabhupada:
The Mayavadi sannyasis, they carry ekadanda, one stick. As we Vaisnava sannyasis carry 3 danda, or three sticks, combined together. The one stick is the symbol of understanding oneness. The monists only accept chin matra, there is only one spirit soul; they do not understand the varieties of the spiritual world. And so far our three sticks are concerned, we take it for granted that we have dedicated our life, for Krishna's service in 3 ways, namely, in our body, in our mind, and in our words.
The connection between the Tridanda and the three ways come from this chapter of the Manu Smriti:
That man is called a (true) tridandin in whose mind these three, the control over his speech (vagdanda), the control over his thoughts (manodanda), and the control over his body (kayadanda), are firmly fixed.
EDIT: The white flag-looking thing is called the Jala Pavitram, and as its name suggests it's used to purify water by filtering. It's mentioned in this verse of the Parashara Smriti:
tatra kutichaka nama putradibhih kutim karayitva kama-krodha-lobha-moha-mada-matsaryadin-parityaja vidhivat-sannyasam kritva tridanda jala-pavitra kashaya-vastra-dharanih
Those sannyasis known as kuticaka are those who request their sons to build them a cottage; they give up lust, anger greed, illusion, madness and envy etc. They renounce the world thus and carry a tridanda, a water-strainer and wear cloth dyed the colour of kashaya.
And here is what this section of the Baudhayana Dharma Sutras says:
With the three means of punishment, (viz.) words, thoughts, and acts, he shall not injure created beings. He shall carry a cloth for straining water for the sake of purification. He shall perform the necessary purifications with water which has been taken out (of a well or tank) and has been strained.
The Jala Pavitram is also used for purifying air during Pranayamam as described in this web page.
In a recent interview to a Telugu news channel, Chinna Jeeyar Swami clarifies why he carries those sticks and the white cloth. Although he's a tridaṇḍi, he carries 4 sticks. Each stick represents a tattva (principle):
The white cloth is called a jala-pavitram used to filter and purify water before drinking. They also have a restriction to drink only water that flows so they don't take bottled water.
He doesn't explain why the sticks are sometimes covered.
The meaning of tridandi is one who does not demand respect for one’s self. He is respectful to others, he is tolerant and he glorifies Hari. The Vaisnava is a demigod, yet he does not consider himself as a demigod. A tridandi means nirasir-nirnamas-kriyah – a tridandi does not bless anybody nor does he accept any respects from anyone. But if one does not offer his respects to a tridandi and does not accept his blessing of krsne matirastu (‘May you always remember Krsna’), one must fast as atonement.
