This great prayer is addressed to the Lord presiding in temple at Thiru vehaa, which is near Kanchi puram. This lord is also called Yathokthakari (He who obeys what is told to him). There is an interesting story about this.
Once Lord Brahma wanted to perform Aswamedha (horse sacrifice) Yagna in the hill near Kanchi called Hastha Giri. Since his consort Saraswathi did not want him to do it, he took help from others. So the enraged goddess came as river Veghavathi(fast moving river) and wanted to destroy the Yaga. But lord Vishnu had decided to help Lord Brahma to do this sacrifice and so he assumed the form of a dam across Vegahvathi and stopped the waters. Vegha Sethu is that dam. This prayer is addressed to Lord in the form of this dam.
He is called Yathokthakari because once when one of his great devotees wanted him, he relocated from Kanchipuram to Thiru Veha, with his folded bed(Adhi Sesha)
॥ Swami Vedantha Desikan Vega Sethu Stotram and Meaning ॥
Sriman Venkatanatharya kavi tharkika Kesari,
Vedanthacharya varyoh may sannidatham hrudhi
Venktanatha the lion among poets and debate,
Who is a great scholar of Vedantha,
Keeps the Lord always in his mind.
Yekam Vegavathi madhye,
Hasthishaile cha drusyathe,
Upaya phala bhavena,
Vyaktham param maha. ॥ 1 ॥
Once in the middle of river Vegavathi,
He himself in his own great divine form.
Was seen from the Hasthi mountain,
As a result of the means to find solution.
Eeshte gamayithum parameswara sethu bhangura,
Yatra saraswatham srotho visramyathi vishrungalam. ॥ 2 ॥
He went there as the Lord in the form of bridge to stop,
The flood of Saraswathi and reposed there for chaining it.
Jayathi jagadekha sethu,Vegavathimadhyalakshitho deva,
prasamayathi Ya: prajaanaam prathithaan samsarajaladhikallolan ॥ 3 ॥
Victory to the only bridge of the world,
Which was really God in the middle of Vegavathi,
And was well known as the bridge to cross
The sea of Samsara full of tumultuous waves.
Vibhathu neh chatesai Vishnusethu:
vegaapagaa vega vigAtha hetu,
Ambhoja Yonehr yadhujnamaaseeth
Abhanga Rakshaa hayamedha Dheekshaa. ॥ 4 ॥
Let the glory of the Vishnu as a bridge brighten,
For it stops the torrential flow of Vegavathi,
With a view to protect the Aswamedha Yaga,
Of the son of the Lord who was born out of lotus.
Chaturaanana saptha tanthu goptha,
saritham Vegavatheemasou nirundhan,
Paripushayathi mangalaani pumsam
bhagavaan bhakthimataam Yathokthakaaree. ॥ 5 ॥
The four faced God with the seven strings,
Was helped by stoppage of the flow of the river Vega vathi
By the God Yathokthakari*, who would bless the devotee,
who salutes him with all that is good.
*The God who would do as he was told.
Sriman pitamaha vadhu paricharyamana:
Setheh bhujanga sayaneh sa mahabhujanga,
Pratyadisanthi bhava samcharanm prajanam
bhakthanuganthuriha yasya gatagatAni. ॥ 6 ॥
The wife of Lord Brahma is doing service,
To Lord Vishnu,who sleeps on the bridge of Adhi sesha,
And who looks after his devotees in very many ways,
As he blesses them by taking care of all the events in their life.
Prasamitha hayamedha vyaapadam padma Yoneh,
sritha jana paratantram sesha bhogeh sayaanam,
Saranamupagathaa: sma: saantha: Nisseshadhosham,
satamanisethum saasvatham vegavatyaa. ॥ 7 ॥
For protecting the Aswa Medha sacrifice carried out by Brahma,
The Lord rests on the snake Adisesha in the middle of Vegavathi,
On the bridge created by hundreds of bluish stones,
And this lord lets us approach him with absolute surrender.
Saranamupagathaanaam soyamaadesakari
samayathi parithaapam sammukha: sarva jantho,
Sataguna pariNama: sannidhaou yasya nityam
vara vitarana bhuma Varanadhreesvarasya . ॥ 8 ॥
He obeys the orders of those who surrender to him,
And puts an end to their sufferings as also all those near to them,
And in front of the place where he is present, he grants boons daily,
But In this world,as Lord Varada he grants hundred folds more than that.
Kanchee Bhagyam kamala nilaya chetaso abhishta siddhi,
kalyananam nidhiravikala: kopi karunyarasi,
Punyanam na: parinathirasou bhooshayan bhogisayam
vegaasethurjayathi vipulo viswarakshaikahetu . ॥ 9 ॥
Due to the luck of Kanchi,To fulfill the wishes of Goddess Lakshmi,
Who is merciful enough to show the right way to the boon of salvation,
The lord who decorates Adhi Sesha appears as a ripened fruit,
At Vega sethu so that, the intention of her saving the world is broadened.
Phala Sruthi
Vegasethoridam stotram Venkatesaena nirmitham,
Yeh patanteh janastheshAm yathoktham kuruteh Hari.
If this Vega Sethu stotra composed by Venkatesa,
Is read,. it would lead to fulfillment of wishes by Lord Vishnu.
Salutation to the author
Kavi thankkika simhaaya, kalyani guna shaline,
SRimath Venkatesaya Vedantha Gurava Mama
Our salutations to Sri Venkatesa, the teacher of Vedantha,
Who is a lion among poets engaged in debate and,
Is the store house of all good qualities.
Srimath Nigamantha Maha Desikaya Nama
Our Salutations to the great teacher Vedantha Desika.